Surface effects on nonlinear vibration of embedded functionally graded nanoplates via higher order shear deformation plate theory

In this paper, free vibration behavior of functionally nanoplate resting on a Pasternak linear elastic foundation is investigated. The study is based on third-order shear deformation plate theory with small scale effects and von Karman nonlinearity, in conjunction with Gurtin–Murdoch surface continuum theory. It is assumed that functionally graded (FG) material distribution varies continuously in the thickness direction as a power law function and the effective material properties are calculated by the use of Mori–Tanaka homogenization scheme. The governing and boundary equations, derived using Hamilton's principle are solved through extending the generalized differential quadrature method. Finally, the effects of power-law distribution, nonlocal parameter, nondimensional thickness, aspect of the plate, and surface parameters on the natural frequencies of FG rectangular nanoplates for different boundary conditions are investigated.     

Nonlinear analysis of non-uniform beams on nonlinear elastic foundation

In this paper a boundary integral equation solution to the nonlinear problem of non-uniform beams resting on a nonlinear triparametric elastic foundation is presented, which permits also the treatment of nonlinear boundary conditions. The nonlinear subgrade model which describes the foundation includes the linear and nonlinear Winkler (normal) parameters and the linear Pasternak (shear) foundation parameter. The governing equations are derived in terms of the displacements for nonlinear analysis in the deformed configuration and for linear analysis in the undeformed one. Moreover, as the cross-sectional properties of the beam vary along its axis, the resulting coupled nonlinear differential equations have variable coefficients which complicate the mathematical problem even more. Their solution is achieved using the analog equation method of Katsikadelis. Several beams are analyzed under various boundary conditions and load distributions, which illustrate the method and demonstrate its efficiency and accuracy. Finally, useful conclusions are drawn from the investigation of the nonlinear response of non-uniform beams resting on nonlinear elastic foundation.     

经典边界条件黏弹性Pasternak地基上Bernoulli-Euler梁横向自振特性分析

自振特性在结构的动力分析中具有重要的意义。将回传射线矩阵法(MRRM)推广到地基梁自振特性的研究中,通过节点力平衡和位移协调方程及对偶局部坐标系下单元相位关系,建立两端简支、两端自由、两端固支、简支-自由、简支-固支及固支-自由这六种边界条件下黏弹性Pasternak地基上的Bernoulli-Euler梁的回传射线矩阵,进而得到其频率方程。根据单一局部坐标系下的边界条件,推导出模态函数解析表达式,进一步根据正交归一化条件求解模态函数表达式中的未知参数。通过具体算例验证了回传射线矩阵法求解的正确性,并对不同边界条件下的自振频率、衰减系数及模态函数进行了分析。为黏弹性地基梁的振动特性研究提供理论基础。     

Bending analysis of FG viscoelastic sandwich beams with elastic cores resting on Pasternak’s elastic foundations

The investigation of bending response of a simply supported functionally graded (FG) viscoelastic sandwich beam with elastic core resting on Pasternak’s elastic foundations is presented. The faces of the sandwich beam are made of FG viscoelastic material while the core is still elastic. Material properties are graded from the elastic interfaces through the viscoelastic faces of the beam. The elastic parameters of the faces are considered to be varying according to a power-law distribution in terms of the volume fraction of the constituent. The interaction between the beam and the foundations is included in the formulation. Numerical results for deflections and stresses obtained using the refined sinusoidal shear deformation beam theory are compared with those obtained using the simple sinusoidal shear deformation beam theory, higher- and first-order shear deformation beam theories. The effects due to material distribution, span-to-thickness ratio, foundation stiffness and time parameter on the deflection and stresses are investigated.     

Differential quadrature method for nonlocal nonlinear vibration analysis of a boron nitride nanotube using sinusoidal shear deformation theory

This article presents a nonlocal sinusoidal shear deformation beam theory (SDBT) for the nonlinear vibration of single-walled boron nitride nanotubes (SWBNNTs). The surrounding elastic medium is simulated based on nonlinear Pasternak foundation. Based on the nonlocal differential constitutive relations of Eringen, the equations of motion of the SWBNNTs are derived using Hamilton's principle. Differential quadrature method (DQM) for the nonlinear frequency is presented, and the obtained results are compared with those predicted by the nonlocal Timoshenko beam theory (TBT). The effects of nonlocal parameter, vibrational modes, length, and elastic medium on the nonlinear frequency of SWBNNTs are considered.     

A general Fourier formulation for vibration analysis of functionally graded sandwich beams with arbitrary boundary condition and resting on elastic foundations

Free vibration analysis of functionally graded sandwich beams with general boundary conditions and resting on a Pasternak elastic foundation is presented by using strong form formulation based on modified Fourier series. Two types of common sandwich beams, namely beams with functionally graded face sheets and isotropic core and beams with isotropic face sheets and functionally graded core, are considered. The bilayered and single-layered functionally graded beams are obtained as special cases of sandwich beams. The effective material properties of functionally graded materials are assumed to vary continuously in the thickness direction according to power-law distributions in terms of volume fraction of constituents and are estimated by Voigt model and Mori–Tanaka scheme. Based on the first-order shear deformation theory, the governing equations and boundary conditions can be obtained by Hamilton’s principle and can be solved using the modified Fourier series method which consists of the standard Fourier cosine series and several supplemented functions. A variety of numerical examples are presented to demonstrate the convergence, reliability and accuracy of the present method. Numerous new vibration results for functionally graded sandwich beams with general boundary conditions and resting on elastic foundations are given. The influence of the power-law indices and foundation parameters on the frequencies of the sandwich beams is also investigated.     

黏弹性三参数地基梁横向自由振动

将复模态方法推广至地基梁系统的振动分析中,研究了三参数描述的黏弹性Pasternak地基梁的横向振动特征,得到不同边界条件下的频率方程的近似解析式以及模态函数表达式。采用数值方法近似求解复模态分析得到的超越方程,并运用微分求积方法数值加以验证。通过具体算例,分析了边界条件、刚度系数以及地基黏性系数等对固有频率和模态函数的影响。研究结果表明,微分求积法的数值解与复模态的近似解析解吻合的很好。     

Axisymmetric free and forced vibrations of initially stressed circular nanoplates embedded in an elastic medium

As a first endeavor, the axisymmetric free and forced vibrations of circular single- and double-layered nanoplates under initial in-plane radial stresses and embedded in an elastic medium are investigated. The governing equations are derived by decoupling the nonlocal constitutive equations of the Eringen theory in polar coordinates in conjunction with the classical plate theory. The elastic medium is modeled as a two-parameter elastic foundation (Pasternak type). Galerkin’s method is employed to solve the resulting equation for vibration frequencies and dynamic response. The effects of small scale together with the other parameters such as initial in-plane load, Winkler and shear elastic foundation coefficients and the radius of the nanoplate are investigated. It is shown that the corresponding natural frequencies obtained by nonlocal elasticity theory are very different from those predicted by classical elasticity theory when the radius of the nanoplate is less than an approximate limit value.     

Free vibration analysis of functionally graded plates resting on Winkler–Pasternak elastic foundations using a new shear deformation theory

Free vibration analysis of simply supported functionally graded plates (FGP) resting on a Winkler–Pasternak elastic foundation are examined by a new higher shear deformation theory in this paper. Present theory exactly satisfies stress boundary conditions on the top and the bottom of the plate. The material properties change continuously through the thickness of the plate, which can vary according to power law, exponentially or any other formulations in this direction. The equation of motion for FG rectangular plates resting on elastic foundation is obtained through Hamilton’s principle. The closed form solutions are obtained by using Navier technique, and then fundamental frequencies are found by solving the results of eigenvalue problems. The numerical results obtained through the present analysis for free vibration of functionally graded plates on elastic foundation are presented, and compared with the ones available in the literature.     

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.     

Influences of elastic foundations and shear deformations on the buckling behavior of functionally graded material truncated conical shells under axial compression

This article presents an investigation on the buckling of functionally graded (FG) truncated conical shells under an axial load resting on elastic foundations within the shear deformation theory (SDT). The governing equations are solved using the Galerkin method, and the closed-form solution of the axial buckling load for FG conical shells on elastic foundations within the SDT is obtained. Various numerical examples are presented and discussed to verify the accuracy of the closed-form solution in predicting dimensionless buckling loads for FG conical shells on the Winkler–Pasternak elastic foundations within the SDT.     

Vibration of double-walled carbon nanotubes coupled by temperature-dependent medium under a moving nanoparticle with multi physical fields

A numerical model on nonlinear vibration of double-walled carbon nanotubes (DWCNTs) subjected to a moving nanoparticle and multi physical fields is proposed. DWCNTs are considered with the kinematic assumption of Euler–Bernoulli beam theory. The surrounding elastic substrate is simulated as Pasternak foundation, which is assumed to be temperature-dependent. Hamilton's principle, incremental harmonic balanced method, Galerkin, and time integration method with direct iteration are employed to establish the equations of motion of zigzag DWCNTs. The study reveals that for the weak van der Waals forces, DWCNTs have the positive and the negative deflections as if it vibrates under a moving nanoparticle.     

Vibration of nonclassical shear beams with Winkler-Pasternak-type restraint

Transverse vibration of the shear beams containing rotary inertia and with a two-parameter elastic foundation is studied. Using asymptotic analysis of Timoshenko beam theory, we derive explicit characteristic equations of the nonclassical shear beams with Winkler-Pasternak elastic restraint and with both ends linked to translational and rotational springs. The condition of the nonclassical shear beams reducing to the classical ones is found. Natural frequencies of the nonclassical modes are evaluated for free- and pinned-elastically restrained shear beams with or without bracing. The influences of elastic restraint stiffness and rotary inertia on the natural frequencies are discussed. Some extreme cases can be recovered from the present. The obtained results are helpful in the design of a tall frame building.     

Out-of-plane free vibration of functionally graded circular curved beams in thermal environment

A first known formulation for the out-of-plane free vibration analysis of functionally graded (FG) circular curved beams in thermal environment is presented. The formulation is based on the first order shear deformation theory (FSDT), which includes the effects of shear deformation and rotary inertia due to both torsional and flexural vibrations. The material properties are assumed to be temperature dependent and graded in the direction normal to the plane of the beam curvature. The equations of motion and the related boundary conditions, which include the effects of initial thermal stresses, are derived using the Hamilton’s principle. Differential quadrature method (DQM), as an efficient and accurate numerical method, is adopted to solve the thermoelastic equilibrium equations and the equations of motion. The formulations are validated by comparing the results, in the limit cases, with the available solutions in the literature for isotropic circular curved beams. In addition, for FG circular curved beams with soft simply supported edges, the results are compared with the obtained exact solutions. Then, the effects of temperature rise, boundary conditions, material and geometrical parameters on the natural frequencies are investigated.     

Non-linear active control of FG beams in thermal environments subjected to blast loads with integrated FGP sensor/actuator layers

Non-linear active control of dynamic response of functionally graded (FG) beams with rectangular cross-section in thermal environments exposed to blast loadings is presented. Two FG piezoelectric layers are bonded to the beam surfaces to act as sensor and actuator. Non-linear equations of motion of the smart beam are derived based on the first-order shear deformation theory and the von Karman geometrical non-linearity. Constant velocity feedback algorithm is used to control the dynamic response of the FG beam actively through closed loop control. The generalized differential quadrature method together with the Newmark-beta scheme is utilized to solve the non-linear partial differential equations in spatial and time domains. The resulted non-linear algebraic equations are then solved using the modified Newton–Raphson method. A detailed analysis of the influence of the geometric non-linearity, material parameters and temperature field on the active vibration control of FG beams subjected to various impulsive loads is carried out.     

Vibration of non-ideal simply supported laminated plate on an elastic foundation subjected to in-plane stresses

In this paper, the effect of non-ideal boundary conditions and initial stresses on the vibration of laminated plates on Pasternak foundation is studied. The plate has simply supported boundary conditions and is assumed that one of the edges of the plate allows a small non-zero deflection and moment. The initial stresses are due to in-plane loads. The vibration problem is solved analytically using the Lindstedt–Poincare perturbation technique. So the frequencies and mode shapes of the plate with non-ideal boundary condition is extracted by considering the Pasternak foundation and in-plane stresses. The results of finite element simulation, using ANSYS software, are presented and compared with the analytical solution. The effect of various parameters like stiffness of foundation, boundary conditions and in-plane stresses on the vibration of the plate is discussed. Dependency of non-ideal boundary conditions on the aspect ratio of the plate for changing the frequencies of vibrations is presented. The relation between the shear modulus of elastic foundation and the frequencies of the plate is investigated.     

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.     

Vibration and buckling analysis of functionally graded sandwich beams by a new higher-order shear deformation theory

This paper proposes a new higher-order shear deformation theory for buckling and free vibration analysis of isotropic and functionally graded (FG) sandwich beams. The present theory accounts a new hyperbolic distribution of transverse shear stress and satisfies the traction free boundary conditions. Equations of motion are derived from Lagrange's equations. Analytical solutions are presented for the isotropic and FG sandwich beams with various boundary conditions. Numerical results for natural frequencies and critical buckling loads obtained using the present theory are compared with those obtained using the higher and first-order shear deformation beam theories. Effects of the boundary conditions, power-law index, span-to-depth ratio and skin-core-skin thickness ratios on the critical buckling loads and natural frequencies of the FG beams are discussed.     

Nonlinear dynamical response of embedded fluid-conveyed micro-tube reinforced by BNNTs

In the present study, nonlinear dynamical behavior and stability of an embedded fluid conveying smart composite micro-tube under imposed electric potential and thermal loadings have been investigated. The composite matrix is the poly-vinylidene fluoride (PVDF) reinforced by double-walled boron nitride nanotubes (DWBNNTs). Composite structure is modeled based on piezoelectric fiber reinforced composite (PFRC) theory and a representative volume element has been considered for predicting the elastic, piezoelectric, dielectric and thermal properties of the smart composite tube. The fluid flow is assumed to be inviscid, irrotational and incompressible. Formulation presented here is based on Euler–Bernoulli beam model with von-Kármán geometric nonlinearity and nonlocal elasticity theory. The interactions between smart composite micro-tube and surrounding elastic media are simulated by Pasternak foundation model. The discretized governing equations of motion are directly obtained by minimizing the energy of the system. As a result, the eigen-values and eigenvectors (mode shapes) are to be obtained by the state-space matrix which is then solved by an iterative method to obtain nonlinear frequencies of smart composite tube. The results significantly show that imposing positive electric potential increases nonlinear stability of the system. In addition, it is concluded that applying electric and thermal loadings can be used as well as controlling parameters to improving stability of the smart composite micro-tube.     

Forced vibration of sinusoidal FG nanobeams resting on hybrid Kerr foundation in hygro-thermal environments

Size-dependent forced vibration behavior of functionally graded (FG) nanobeams subjected to an in-plane hygro-thermal loading and lateral concentrated and uniform dynamic loads is investigated via a higher-order refined beam theory, which captures shear deformation influences needless of any shear correction factor. The nanobeam is in contact with a three-parameter Kerr foundation consisting of upper and lower spring layers as well as a shear layer. Hygro-thermo-elastic material properties of the nanobeam are described via power-law distribution considering exact position of the neutral axis. Through nonlocal elasticity theory of Eringen and Hamilton's principle, the governing equations of higher-order FG nanobeams on Kerr foundation under dynamic loading are derived. These equations are solved for simply-supported and clamped-clamped boundary conditions. A detailed parametric study is performed to show the importance of moisture concentration rise, temperature rise, material composition, nonlocality, Kerr foundation parameters, and boundary conditions on forced vibration characteristics and resonance frequencies of FG nanobeams. As a consequence, Kerr foundation parameters lead to a significant delay in the occurrence of resonance frequencies.