Thermal stress and magnetic effects on nonlinear vibration of nanobeams embedded in nonlinear elastic medium

Abstract

Nonlinear vibration of nanobeams embedded in the linear and nonlinear elastic materials under magnetic and temperature effects is investigated in this study. Von Karman’s strain–displacement relation is applied to a nonlocal Euler–Bernoulli beam model. Equation of motion is derived using Hamilton’s principle. Galerkin’s method is applied to decompose the nonlinear partial differential equation into a nonlinear ordinary differential equation (NODE). The NODE is solved using He’s method. The nanobeams are embedded in the Winkler, Pasternak, and nonlinear elastic media. The effects of low and high temperatures, nonlocal parameter, magnetic force, amplitude, and linear and nonlinear elastic materials are examined.     

Thermal Buckling Analysis of Embedded Single-Walled Carbon Nanotubes with Arbitrary Boundary Conditions Using the Nonlocal Timoshenko Beam Theory

In this paper, the thermal buckling behavior of single-walled carbon nanotubes (SWCNTs) embedded in an elastic medium is studied. To this end, the SWCNTs are modeled based on the nonlocal Timoshenko beam theory into which the effect of the elastic medium is incorporated The generalized differential quadrature (GDQ) method is employed to discretize the governing differential equations and to consider different commonly used boundary conditions (BCs). For simply supported BCs, the results obtained from the present analysis are compared with the ones from the exact solution and an excellent agreement has been achieved. The effects of the aspect ratio, nonlocal parameter and the Winkler parameter on the dimensionless critical buckling temperature are carefully investigated.     

内燃机轴系扭振响应的扭转弹性波计算方法研究

以工程中常见的阶梯轴为研究对象,建立了连续轴模型中沿轴向传播的扭振弹性波的波动方程组,结合方程组的初始条件和边界条件及波动方程解的行波表达式,导出了各轴段扭转弹性波传播的解析表达式,利用数值迭代方法可同时计及各谐波的综合作用,要精确计算共振及非共振工况的强迫振动响应,使计算过程与扭振实测分析过程完全对应。     

Analytical Solution for Thermomechanical Vibration of Double-Viscoelastic Nanoplate-Systems Made of Functionally Graded Materials

In this article, based on the nonlocal elasticity theory of Eringen, dynamic characteristics of a double-FGM viscoelastic nanoplates-system subjected to temperature change with considering surface effects (surface elasticity, tension and density) is studied. Two Kirchhoff nanoplates are coupled by an internal Kelvin–Voigt viscoelastic medium and also are limited to the external Pasternak elastic foundation. The material properties of the simply supported functionally graded nanoplates are assumed to follow power law distribution in the thickness direction. The governing equations of motion for three cases (out-of-phase vibration, in-phase vibration and one nanoplate fixed) are derived from Hamilton's principle. The analytical approach is employed to determine explicit closed-form expression for complex natural frequencies of the system. Numerical results are presented to show variations of the frequency of double-FGM viscoelastic nanoplates corresponding to various values of the nonlocal parameter, temperature change, power law index, aspect ratio and transverse and shear stiffness coefficients of the Pasternak elastic foundation. Moreover, influence of higher order modes, viscoelastic structural damping and damping coefficient of the viscoelastic medium on vibration characteristics are investigated. Numerical results show that natural frequency is greatly influenced by surface elastic modulus and residual surface stress.     

Hygrothermal effects on static stability of embedded single-layer graphene sheets based on nonlocal strain gradient elasticity theory

Abstract

This article develops a nonlocal strain gradient plate model for buckling analysis of graphene sheets under hygrothermal environments. For more accurate analysis of graphene sheets, the proposed theory contains two scale parameters related to the nonlocal and strain gradient effects. Graphene sheet is modeled via a two-variable shear deformation plate theory needless of shear correction factors. Governing equations of a nonlocal strain gradient graphene sheet on elastic substrate are derived via Hamilton’s principle. Galerkin’s method is implemented to solve the governing equations for different boundary conditions. Effects of different factors such as moisture concentration rise, temperature rise, nonlocal parameter, length scale parameter, elastic foundation and geometrical parameters on buckling characteristics a graphene sheets are examined.     

Free vibration analysis of a nonlocal thermoelastic hollow cylinder with diffusion

Abstract

In this article, the constitutive relations and the governing equations are derived for nonlocal thermoelastic solid in the presence of diffusion. The free vibration of a thermoelastic diffusive cylinder is investigated within the framework of the above newly derived model. Time-harmonic vibration is used to transform the governing equations into a system of ordinary differential equations. The frequency equation is taken under investigation for the survival of a range of possible modes in compact form for traction-free thermal boundary conditions: thermally insulated and isothermal boundary conditions. To explore the vibration analysis from frequency equations, we apply a numerical iteration technique for generating numerical data by taking assistance of the Matlab software. The numerically computed and simulated results for the frequency shift, natural frequency, and the thermoelastic damping are presented graphically. The effect of nonlocality on the above quantities is observed and shown graphically.     

Through-the-length temperature distribution effects on thermal vibration analysis of nonlocal strain-gradient axially graded nanobeams subjected to nonuniform magnetic field

Thermomechanical vibration analysis of axially functionally graded (AFG) nanobeams under nonuniform longitudinal magnetic field is investigated based on nonlocal strain-gradient theory. This theory contains two scale parameters for modeling of size-dependent behavior of AFG nanobeam accurately. This theory takes into account both nonlocal stress field and strain-gradient effects on the response of nanostructures. The nanobeam is subjected to uniform and linear through-the-length temperature distributions. A power-law model is used to describe the distribution of temperature-dependent material properties along the axial direction. A Galerkin-based solution technique is implemented to solve the governing equation obtained from Hamilton’s principle. Natural frequencies of functionally graded nanobeam are verified with those of previous articles. It is shown that vibration frequencies of AFG nanobeams are significantly influenced by temperature rise, power-law index, nonlocal parameter, length-scale parameter, magnetic field intensity, and boundary conditions.     

不同边界条件下吸力筒导管架式海上风电振动特性分析

  目的  吸力筒导管架是一种具备许多优点的新型海上风电基础,其振动特性与传统基础有着明显的不同。正确处理无限域地基的边界条件对准确分析振动问题至关重要,故对不同边界条件下吸力筒导管架式海上风电的振动特性进行分析。  方法  以某吸力筒导管架式海上风电为分析对象,在ANSYS APDL中建立结构与土体耦合的全三维有限元模型,研究边界条件对自振频率的影响并对相关参数进行敏感性分析。  结果  不同边界条件对整机固有频率影响较大,改变边界条件对结构轴向刚度和扭转刚度的影响大于对弯曲刚度的影响,整机固有频率随着粘弹性边界弹簧刚度、土体弹性模量和筒土摩擦系数的增大而增大。  结论  分析了影响整机固有频率计算精度的相关因素,可为工程计算提供参考。     

Exact Solution for Nonlinear Stability of Piezoelectric FGM Timoshenko Beams Under Thermo-Electrical Loads

In this article an exact solution is presented for the nonlinear response of hybrid functionally graded material (FGM) Timoshenko beams subjected to simultaneous action of thermal and electrical loads. Properties of the FGM media are graded across the thickness based on a power law form. Employing the Timoshenko beam theory and mid-surface based formulation in conjunction with the von-Karman strain-displacement relations, the three non-linear equilibrium equations along with the associated boundary conditions are obtained. The resulting equations are then uncoupled in a reasonable manner. An exact closed-form solution is presented to trace the load-deflection path for the clamped and simply supported beams. It is shown that the behavior of these two types of boundary conditions are totally different since the response of FGM clamped beam is of the bifurcation-type while the load-deflection path of FGM simply supported beams is unique and stable. This feature is detectable through the temperature-deflection or force-temperature paths. Numerical results are presented to investigate the effects of various involved parameters.     

Thermoelastic Response of Cross-Ply Laminated Shells Based on a Rigorous Shell Theory

Thermal deformations in cross-ply laminated shells are investigated. The state space approach is used to generate exact solutions for the thermoelastic response of cross-ply spherical, cylindrical and doubly curved shells for various boundary conditions and subjected to general temperature field. The shells possess two parallel edges simply supported and the remaining ones having any possible combination of boundary conditions: free, clamped or simply supported. A rigorous thick deep first order shear deformation shell theory is used in the analysis. Deflections are computed for shells with various boundary conditions undergoing uniform and linearly varying temperature through the thickness. The exact solutions for thermal deflections can be used as benchmarks for approximate solutions such as Rayleigh-Ritz and finite element methods.     

Asymmetric thermal buckling of annular plates on a partial elastic foundation

In this investigation, the asymmetrical buckling behavior of isotropic homogeneous annular plates resting on a partial Winkler-type elastic foundation under uniform temperature elevation is investigated. First-order shear deformation plate theory is used to obtain the governing equations and the associated boundary conditions. Prebuckling deformations and stresses of the plate are obtained under the solution of a plane stress formulation, neglecting the rotations and lateral deflection. Applying the adjacent equilibrium criterion, the linearized stability equations are obtained. The governing equations are divided into two sets. The first set, which is associated with the in-contact region, and the second set, which is related to contact-less region. The resulting equations are solved using a hybrid method, including the analytical trigonometric functions through the circumferential direction and generalized differential quadratures method through the radial direction. The resulting system of eigenvalue problem is solved to obtain the critical conditions of the plate and the associated circumferential mode number. Benchmark results are given in tabular and graphical presentations for combinations of simply supported and clamped types of boundary conditions. Numerical results are given to explore the effects of elastic foundation, foundation radius, plate thickness, plate hole size, and the boundary conditions.     

Thermomechanical nonlinear analysis of axially compressed carbon nanotube-reinforced composite cylindrical panels resting on elastic foundations with tangentially restrained edges

Buckling, postbuckling, and nonlinear responses of composite cylindrical panels reinforced by single-walled carbon nanotubes (CNTs), supported by an elastic foundation, exposed to elevated temperature and axially compressed by uniform load are investigated in this article. Distribution of CNTs is uniform or graded in the thickness direction and the effective properties of CNT-reinforced composite are assumed to be temperature dependent, and are estimated by extended rule of mixture through a micromechanical model. Governing equations are established based on thin shell theory taking von Kármán–Donnell nonlinearity, initial geometrical imperfection, Pasternak-type elastic foundation and tangential elastic constraints of boundary edges into consideration. Approximate solutions of deflection and stress functions are assumed to satisfy simply supported boundary conditions, and Galerkin method is applied to derive explicit expressions of load–deflection relation from which critical buckling loads can be obtained. Unlike works in the literature, the present study accounts for elasticity of tangential restraint of two unloaded straight edges in model of cylindrical panel. The study also gives conditions for which bifurcation type buckling response can occur and novel findings in numerical examples.     

ON THE PROPAGATION OF THERMOELASTIC WAVES IN MEDIA WITH LONG-RANGE COHESIVE FORCES

Abstract

After establishing the fundamental equations of nonlocal coupled thermoelasticity in Fourier space, nonlocal longitudinal thermoelastic waves in an infinite space are analyzed. Identification of the elastic dispersion equation obtained with its counterpart derived in lattice dynamics leads to values of the nonlocal elastic moduli. Similar identification of the velocity of thermal waves with its counterpart in second sound theory yields the values of the nonlocal thermal moduli. A numerical example involving solid helium is given.     

Geometrically nonlinear dynamic analysis of doubly curved isotropic shells resting on elastic foundation by a combination of harmonic differential quadrature-finite difference methods

The nonlinear dynamic response of doubly curved shallow shells resting on Winkler–Pasternak elastic foundation has been studied for step and sinusoidal loadings. Dynamic analogues of Von Karman–Donnel type shell equations are used. Clamped immovable and simply supported immovable boundary conditions are considered. The governing nonlinear partial differential equations of the shell are discretized in space and time domains using the harmonic differential quadrature (HDQ) and finite differences (FD) methods, respectively. The accuracy of the proposed HDQ-FD coupled methodology is demonstrated by numerical examples. The shear parameter G of the Pasternak foundation and the stiffness parameter K of the Winkler foundation have been found to have a significant influence on the dynamic response of the shell. It is concluded from the present study that the HDQ-FD methodolgy is a simple, efficient, and accurate method for the nonlinear analysis of doubly curved shallow shells resting on two-parameter elastic foundation.     

Nonlocal Flugge Shell Model for Thermal Buckling of Multi-Walled Carbon Nanotubes with Layerwise Boundary Conditions

Presented herein is the thermal buckling analysis of multi-walled carbon nanotubes on the basis of nonlocal Flugge shell model capturing small scale effects. Based upon the continuum mechanics, a multiple-shell model is adopted in which the nested tubes are coupled with each other through the van der Waals interlayer interaction. The utilized van der Waals model incorporating the interlayer interactions between any two layers, whether adjacent or non-adjacent is curvature dependent. To analytically solve the problem, the Rayleigh–Ritz method was implemented to the variational form equivalent to the Flugge type equations. The present analysis provides the possibility of considering different combinations of layerwise boundary conditions. It is shown that the shell-like thermal buckling is significantly sensitive to the nonlocal parameter variation, whereas the column-like thermal buckling remains unaffected when the nonlocal parameter is varied.     

Thermoelastic-induced vibrations on an elliptical disk with internal heat sources

In this study, integral transform technique is used to investigate the thermally induced vibration of an elliptical disk. The axisymmetric temperature distribution in the disk is determined by conductivity equation and the corresponding initial and boundary conditions using an extended integral transform technique. The problem of thermally induced vibration of the disk with both ends clamped extremes is solved by developing an integral transform for double Laplace differential equation. The thermal moment is derived on the basis of temperature field, whereas maximum normal stresses are derived based on resultant bending moments per unit width. The results are obtained in series form in terms of Mathieu functions, and numerical results are shown in figures.     

MAYSEL'S METHOD FOR A CIRCULAR PLATE UNDER THERMAL/PIEZOELECTRIC LOADING

Maysel's formula is applied to a thin circular plate under axisymmetric thermal or piezoelectric loading. The plate may be traction-free, simply supported, or clamped on its cylindrical boundary. Solutions are obtained in the form of Fourier-Bessel series. Numerical results for displacements and stresses are presented, corresponding to various loading distributions and plate thickness-to-diameter ratios.     

Thermal buckling and postbuckling responses of geometrically imperfect FG porous beams based on physical neutral plane

Abstract

Considering the third-order shear deformation and physical neutral plane theories, thermal postbuckling analysis for functionally graded (FG) porous beam are performed in this research. The cases of shear deformable functionally graded materials (FGM) beams with initial deflection and uniformly distributed porosity are considered. Geometrically imperfect FG porous beams with two different types of immovable boundary conditions as clamped–rolling and clamped–clamped are analyzed. Thermomechanical nonhomogeneous material properties of the FG porous beam are assumed to be temperature and position dependent. FG porous beams are subjected to different types of thermal loads as heat conduction and uniform temperature rise. Heat conduction equation is solved analytically using the polynomial series solution for the one-dimensional condition. The governing equilibrium equations are obtained by applying the virtual displacement principle. Assuming von Kármán type of geometrical nonlinearity, equilibrium equations are nonlinear and are solved using an analytical method. A two-step perturbation technique is used to obtain the thermal buckling and postbuckling responses of FG porous beams. The numerical results are compared with the case of perfect FGM Timoshenko beams without porosity distribution based on the midplane formulation. Parametric studies of the perfect/imperfect FG porous beams for two types of thermal loading and boundary conditions are provided.     

Thermomechanical postbuckling behavior of CNT-reinforced composite sandwich plate models resting on elastic foundations with elastically restrained unloaded edges

Buckling and postbuckling behaviors of two models of sandwich plate reinforced by carbon nanotubes (CNTs) resting on elastic foundations and subjected to uniaxial compressive and thermomechanical loads are investigated in this paper. Material properties of all constituents are assumed to be temperature dependent and effective properties of CNT-reinforced composite layer are determined according to extended rule of mixture. Governing equations are established within the framework of first-order shear deformation theory taking into account von Kármán nonlinearity, initial geometrical imperfection, plate-foundation interaction and tangential elastic constraints of unloaded edges. Three types of loading are considered including uniaxial compression, preexisting thermal load combined with uniaxial compression and preexisting mechanical load combined with thermal load. Approximate analytical solutions are assumed to satisfy simply supported boundary conditions and the Galerkin method is used to derive nonlinear load-deflection relations from which buckling loads and postbuckling equilibrium paths are determined. The most important findings are that tangential constraints of unloaded edges significantly lowers buckling loads and postbuckling load capacity of sandwich plates and, in contrast, buckling loads and postbuckling strength are considerably enhanced as sandwich plate is constructed from CNT-reinforced composite core layer and homogeneous face sheets.     

永磁悬浮支撑的刚性圆柱振子流致振动幅频特性分析

将永磁悬浮支撑与流致振动潮流能转换装置相结合,探索永磁悬浮支撑结构作为弹簧在流致振动潮流能发电领域的应用可行性。为了对比永磁弹簧与金属弹簧所提供恢复力对刚性圆柱振子流致振动的影响,通过ANSYS Maxwell软件计算得到永磁弹簧的磁力,拟合出永磁弹簧的弹性恢复力曲线方程,将弹性恢复力曲线方程代入Star-CCM+计算出刚性圆柱振子在不同流速下的幅频特性,然后将计算结果与金属弹簧系统进行比较。结果显示永磁弹簧支撑振子流致振动幅频特性与金属弹簧的相似,在上端分支能达到最大振幅比,频率随着流速增加总体呈增加趋势。但在计算流速区间内,永磁弹簧支撑振子的振幅比与频率都比金属弹簧支撑振子大,振动性能更优。