Surface effect on the biaxial buckling and free vibration of FGM nanoplate embedded in visco-Pasternak standard linear solid-type of foundation

In this study, nonlocal elasticity theory in conjunction with Gurtin–Murdoch elasticity theory is employed to investigate biaxial buckling and free vibration behavior of nanoplate made of functionally graded material (FGM) and resting on a visco-Pasternak standard linear solid-type of the foundation. The material characteristics of simply supported FGM nanoplates are assumed to be varied continuously as a power law function of the plate thickness. Hamilton’s principle is implemented to derive the non-classical governing equations of motion and related boundary conditions, which analytically solved to obtain the explicit closed-form expression for complex natural frequencies and buckling loads. Finally, attention is focused on considering the influences of various parameters on variation of damped natural frequency and buckling load ratio such as nonlocal parameter, surface effects, geometric parameters, power law index and properties of visco-Pasternak foundation and it is clearly demonstrated that these factors highly affect on vibration and buckling behavior.     

四角点支撑纳米板稳态受迫振动解析解

本文基于非局部弹性理论及辛叠加方法,得到放置在黏弹性介质上四角点支撑矩形纳米板稳态受迫振动问题的解析解．将纳米板受迫振动问题导入哈密顿体系,得到哈密顿控制方程,在无需任何预设函数的情况下可直接对哈密顿控制方程进行求解,得到简支纳米板稳态受迫振动问题在辛空间展开形式的解析解．进而通过边界叠加,可求出四角点支撑纳米板稳态受迫振动的解析解．数值算例中验证了本文应用辛叠加方法得到解析解的准确性,并以石墨烯纳米板为例,分析了非局部参数和黏弹性介质参数对四角点支撑石墨烯纳米板稳态受迫振动的影响．结果表明,非局部参数和黏弹性介质参数的变化会影响石墨烯纳米板的共振频率及共振幅值．     

Bending and vibration of two-dimensional decagonal quasicrystal nanoplates via modified couple-stress theory

Based on the modified couple-stress theory, the three-dimensional(3D)bending deformation and vibration responses of simply-supported and multilayered twodimensional(2D) decagonal quasicrystal(QC) nanoplates are investigated. The surface loading is assumed to be applied on the top surface in the bending analysis, the tractionfree boundary conditions on both the top and bottom surfaces of the nanoplates are used in the free vibration analysis, and a harmonic concentrated point loading is applied o...     

Size-dependent shear buckling response of FGM skew nanoplates modeled via di?erent homogenization schemes

The size e?ects on the shear buckling behaviors of skew nanoplates made of functionally graded materials(FGMs) are presented. The material properties are supposed to be changed uniformly from the ceramic phase to the metal one along the plate thickness. To estimate the associated e?ective material properties, various homogenization schemes including the Reuss model, the Voigt model, the Mori-Tanaka model, and the Hashin-Shtrikman bound model are used. The nonlocal elasticity theory together with the oblique coordinate system is applied to the higher-order shear deformation plate theory to develop a size-dependent plate model for the shear buckling analysis of FGM skew nanoplates. The Ritz method using Gram-Schmidt shape functions is used to solve the size-dependent problem. It is found that the signi?cance of the nonlocality in the reduction of the shear buckling load of an FGM skew nanoplate increases for a higher value of the material property gradient index. Also, by increasing the skew angle, the critical shear buckling load of an FGM skew nanoplate enhances. This pattern becomes a bit less signi?cant for a higher value of the material property gradient index. Furthermore,among various homogenization models, the Voigt and Reuss models in order estimate the overestimated and underestimated shear buckling loads, and the di?erence between them reduces by increasing the aspect ratio of the skew nanoplate.     

Size-dependent shear buckling response of FGM skew nanoplates modeled via different homogenization schemes

The size effects on the shear buckling behaviors of skew nanoplates made of functionally graded materials (FGMs) are presented. The material properties are supposed to be changed uniformly from the ceramic phase to the metal one along the plate thickness. To estimate the associated effective material properties, various homogenization schemes including the Reuss model, the Voigt model, the Mori-Tanaka model, and the Hashin-Shtrikman bound model are used. The nonlocal elasticity theory together with the oblique coordinate system is applied to the higher-order shear deformation plate theory to develop a size-dependent plate model for the shear buckling analysis of FGM skew nanoplates. The Ritz method using Gram-Schmidt shape functions is used to solve the size-dependent problem. It is found that the significance of the nonlocality in the reduction of the shear buckling load of an FGM skew nanoplate increases for a higher value of the material property gradient index. Also, by increasing the skew angle, the critical shear buckling load of an FGM skew nanoplate enhances. This pattern becomes a bit less significant for a higher value of the material property gradient index. Furthermore, among various homogenization models, the Voigt and Reuss models in order estimate the overestimated and underestimated shear buckling loads, and the difference between them reduces by increasing the aspect ratio of the skew nanoplate.     

基于非局部弹性理论的纳米板横向振动

本文利用非局部弹性理论研究了单层石墨烯的纳米板的横向自由振动响应．通过迭代法推导了非局部应力表达,进一步通过哈密顿原理推导了纳米板的控制方程,应用纳维解法得到四边简支纳米板振动固有频率的数值解,并将本文研究结果与已有文献结果进行对比,进一步讨论了小尺寸效应,以及纳米板的三维尺寸和半波数对振动频率的影响．结果表明：非局部效应的存在使得纳米板的等效刚度和固有频率降低;半波数的增加则使得纳米板的固有频率提高．相关分析结果对基于二维纳米材料的新设备的设计和优化具有重要意义．     

Small-scale effects on the buckling of quadrilateral nanoplates based on nonlocal elasticity theory using the Galerkin method

The elastic buckling behavior of quadrilateral single-layered graphene sheets (SLGS) under bi-axial compression is studied employing nonlocal continuum mechanics. Small-scale effects are taken into consideration. The principle of virtual work is employed to derive the governing equations. The Galerkin method in conjunction with the natural coordinates of the nanoplate is used as a basis for the analysis. The buckling load of skew, rhombic, trapezoidal, and rectangular nanoplates considering various geometrical parameters are obtained. It is shown that nonlocal effects are very important in arbitrary quadrilateral graphene sheets and their inclusion results in smaller buckling loads. Also the effects of geometrical parameters such as aspect ratio, angle, and mode number on the buckling load decrease when scale coefficient increases, for all arbitrary quadrilateral SLGS.     

一维六方准晶层合简支梁自由振动与屈曲的精确解

伪Stroh型公式能够将多场耦合材料的控制方程转化为线性特征系统来求解,从而获得多层结构简支边界条件的精确解.本文利用伪Stroh型公式,研究一维六方准晶层合简支梁的自由振动和屈曲问题,通过传递矩阵法,获得准晶层合梁自由振动固有频率与临界屈曲载荷的精确解.通过与已有梁的剪切变形理论结果比较,验证了本文伪Stroh型公式的正确性和有效性.通过数值算例,分析由两种不同准晶材料组成的三明治层合梁的叠层方式、高跨比、层厚比及层数对梁的固有频率、临界屈曲载荷及其模态的影响规律.结果表明,叠层顺序和梁的高跨比、层厚比对准晶层合梁的自由振动固有频率和临界屈曲载荷有很大影响,可通过调整梁的几何尺寸和叠层顺序得到准晶层合梁的最佳固有频率和临界屈曲载荷.本文给出的精确解可为工程上研究准晶梁的各种数值解法和实验方法提供理论参考.     

考虑非局部效应的双层纳米板的磁弹性随机振动

研究了四边简支双层纳米板在外加磁场的作用下的磁弹性随机振动问题．基于非局部弹性理论和板壳 磁弹性理论建立了系统的磁弹性随机振动方程．通过模态分析法对其进行位移响应分析，得到了通入平稳随机 电流时双层纳米板位移响应均值、功率谱密度函数等数字特征．在此基础上，分析了非局部参数、磁场强度、 板厚比等对功率谱密度的影响．结果表明，非局部参数、磁场强度、板厚比等因素的变化会影响系统的振动能 量变化及振动响应带宽分布.     

Buckling and post-buckling analyses of size-dependent piezoelectric nanoplates

This paper attempts to investigate the buckling and post-buckling behaviors of piezoelectric nanoplate based on the nonlocal Mindlin plate model and von Karman geometric nonlinearity. An external electric voltage and a uniform temperature rise are applied on the piezoelectric nanoplate. Both the uniaxial and biaxial mechanical compression forces will be considered in the buckling and post-buckling analysis. By substituting the energy functions into the equation of the minimum total potential energy principle,the governing equations are derived directly, and then discretized through the differential quadrature(DQ) method. The buckling and post-buckling responses of piezoelectric nanoplates are calculated by employing a direct iterative method under different boundary conditions. The numerical results are presented to show the influences of different factors including the nonlocal parameter, electric voltage,and temperature rise on the buckling and post-buckling responses.     

Free vibration of size-dependent magneto-electro-elastic nanoplates based on the nonlocal theory

In this paper, the free vibration of magneto- electro-elastic （MEE） nanoplates is investigated based on the nonlocal theory and Kirchhoff plate theory. The MEE nanoplate is assumed as all edges simply supported rectan gular plate subjected to the biaxial force, external electric potential, external magnetic potential, and temperature rise. By using the Hamilton＇s principle, the governing equations and boundary conditions are derived and then solved analytically to obtain the natural frequencies of MEE nanoplates. A parametric study is presented to examine the effect of the nonlocal parameter, thermo-magneto-electro-mechanical loadings and aspect ratio on the vibration characteristics of MEE nanoplates. It is found that the natural frequency is quite sensitive to the mechanical loading, electric loading and magnetic loading, while it is insensitive to the thermal loading.     

An analytical study of vibration in functionally graded piezoelectric nanoplates: nonlocal strain gradient theory

In this paper, we analytically study vibration of functionally graded piezoelectric(FGP) nanoplates based on the nonlocal strain gradient theory. The top and bottom surfaces of the nanoplate are made of PZT-5 H and PZT-4, respectively. We employ Hamilton's principle and derive the governing differential equations. Then, we use Navier's solution to obtain the natural frequencies of the FGP nanoplate. In the first step, we compare our results with the obtained results for the piezoelectric nanoplates in the previous studies. In the second step, we neglect the piezoelectric effect and compare our results with those obtained for the functionally graded(FG) nanoplates. Finally, the effects of the FG power index, the nonlocal parameter, the aspect ratio, and the lengthto-thickness ratio, and the nanoplate shape on natural frequencies are investigated.     

Application of Ritz functions in buckling analysis of embedded orthotropic circular and elliptical micro/nano-plates based on nonlocal elasticity theory

A continuum model based on the nonlocal theory of elasticity is developed for buckling analysis of embedded orthotropic circular and elliptical micro/nano-plates under uniform in-plane compression. The nanoplate is considered to be rested on two-parameter Winkler-Pasternak elastic foundation. The principle of virtual work is used to derive the governing vibration and stability equations. The weighted residual statements of the equations of motion are performed and the well-known Galerkin method is employed to obtain the nonlocal “Quadratic Functional” for embedded micro/nano-plates. The Ritz functions are taken to form an expression for transverse displacement which satisfies the kinematic boundary conditions. In this way, the entire nanoplate is considered as a single super-continuum element. Employing the Ritz functions eliminates the need for mesh generation and thus large number of degrees of freedom arising in discretization methods such as finite element (FE). The results show obvious dependency of critical buckling loads on the non-locality of the micro/nano elliptical plate, especially, at very small dimensions.     

Thermo-electro-magneto-mechanical bending behavior of size-dependent sandwich piezomagnetic nanoplates

Thermo-electro-magneto-mechanical bending analysis of a sandwich nanoplate is presented in this paper based on Kirchhoff’s plate theory and nonlocal theory. The sandwich nanoplate includes an elastic nano-core and two piezomagnetic face-sheets actuated by applied electric and magnetic potentials. The governing equations for the electro-magneto-mechanical bending are derived in terms of the displacement components and electric and magnetic potentials. Then, the problem is solved analytically by using Navier’s method. A parametric study is presented to show the effects of the nonlocal parameter, temperature rise, applied electric and magnetic potentials on the bending behaviors of sandwich nanoplates for simply-supported boundary conditions. As a main result of study, it is concluded that the deflection decreases as applied electric potential increases and applied magnetic potential decreases. In addition, the increase of nonlocal parameter leads to increase of deflection and maximum electric potential through the thickness direction.     

Free vibration analysis of quadrilateral nanoplates based on nonlocal continuum models using the Galerkin method: the effects of small scale

Free vibration analysis of quadrilateral single-layered graphene sheets (SLGS) is carried out employing nonlocal continuum mechanics. The equations of motion of the nonlocal theory are derived using the principle of virtual work. The Galerkin method in conjunction with the natural coordinates of the nanoplate is used as a basis for the analysis. The non-dimensional natural frequencies of skew, rhombic, trapezoidal and rectangular nanoplates considering various geometrical parameters and mode numbers are obtained and for each case the effects of the small length scale are investigated.     

Modified von Kármán equations for elastic nanoplates with surface tension and surface elasticity

In this paper, modified von Kármán equations are derived for Kirchhoff nanoplates with surface tension and surface tension-induced residual stresses. The simplified Gurtin-Murdoch model which does not contain non-strain displacement gradients in surface stress-strain relations is adopted, so that the von Kármán strain-compatibility equation can be expressed in terms of the stress function and deflection. The modified von Kármán equations derived here are different than the existing related models especially for elastic plates with in-plane movable edges. Unlike the existing models which predict a surface tension-induced tensile pre-stress for an elastic plate with in-plane movable edges, the present model predicts that this tensile pre-stress is actually cancelled by the surface tension-induced residual compressive stress. Our this result is consistent with recent clarification on similar issue for cantilever beams with surface tension, which implies that the existing models have incorrectly predicted an invalid tensile pre-stress for an elastic plate with in-plane movable edges which leads to significant overestimation of postbuckling load and free vibration frequencies. In addition, our numerical examples indicated that surface stresses can moderately increase or decrease postbuckling load and free vibration frequency of Kirchhoff nanoplate with all in-plane movable edges, depending on the surface elasticity parameters and the geometrical dimensions of nanoplates.     

Effects of local thickness defects on the buckling of micro-beam

A buckling model of Timoshenko micro-beam with local thickness defects is established based on a modified gradient elasticity. By introducing the local thickness defects function of the micro-beam, the variable coe-cient differential equations of the buckling problem are obtained with the variational principle. Combining the eigensolution series of the complete micro-beam with the Galerkin method, we obtain the critical load and buckling modes of the micro-beam with defects. The results show that the depth and location of the defect are the main factors affecting the critical load, and the combined effect of boundary conditions and defects can significantly change the buckling mode of the micro-beam. The effect of defect location on buckling is related to the axial gradient of the rotation angle, and defects should be avoided at the maximum axial gradient of the rotation angle. The model and method are also applicable to the static deformation and vibration of the micro-beam.     

Thermal buckling of nanorod based on non-local elasticity theory

The buckling of nanostructures including as a nanobeam, nanorod, and nanotube in a temperature field is investigated based on the non-local elasticity field theory with non-linear strain gradients first proposed by Eringen. New higher-order governing differential equations both in transverse and axial direction for buckling of such nanostructures are derived based on the exact variational principle approach with corresponding higher-order non-local boundary conditions. Based on these new governing equations and boundary conditions, new analytical solutions for some practical examples on buckling of nanostructures are presented and analyzed in detail. Subsequently, the effects of non-local nanoscale and temperature change on critical buckling load are analyzed and discussed. It is observed that those factors have great influence on the critical buckling load of the nanostructures. In particular, the non-local stress very much affects the stiffness of nanostructures and the critical buckling load is significantly increased in the presence of non-local stress. The paper concludes that at low and room temperature the critical buckling load of nanostructures increases with increasing temperature change, while at high temperature the critical buckling load decreases with increasing temperature change. A critical temperature change which causes buckling without external load is also derived and discussed.     

A comprehensive analysis of functionally graded sandwich plates: Part 2—Buckling and free vibration

The sinusoidal shear deformation plate theory, presented in the first part of this paper, is used to study the buckling and free vibration of the simply supported functionally graded sandwich plate. Effects of rotatory inertia are considered. The critical buckling load and the vibration natural frequency are investigated. Some available results for sandwich plates non-symmetric about the mid-plane can be retrieved from the present analysis. The influences of the transverse shear deformation, plate aspect ratio, side-to-thickness ratio and volume fraction distributions are studied. In addition, the effect of the core thickness, relative to the total thickness of the plate, on the critical buckling load and the eigenfrequencies is investigated.     

Nonlocal buckling of embedded magnetoelectroelastic sandwich nanoplate using refined zigzag theory

This paper is concerned with a buckling analysis of an embedded nanoplate integrated with magnetoelectroelastic (MEE) layers based on a nonlocal magnetoelectroelasticity theory. A surrounding elastic medium is simulated by the Pasternak foundation that considers both shear and normal loads. The sandwich nanoplate (SNP) consists of a core that is made of metal and two MEE layers on the upper and lower surfaces of the core made of BaTiO₃/CoFe₂O₄. The refined zigzag theory (RZT) is used to model the SNP subject to both external electric and magnetic potentials. Using an energy method and Hamilton’s principle, the governing motion equations are obtained, and then solved analytically. A detailed parametric study is conducted, concentrating on the combined effects of the small scale parameter, external electric and magnetic loads, thicknesses of MEE layers, mode numbers, and surrounding elastic medium. It is concluded that increasing the small scale parameter decreases the critical buckling loads.