Size-dependent axial buckling analysis of functionally graded circular cylindrical microshells based on the modified strain gradient elasticity theory

In this paper, a size-dependent first-order shear deformable shell model is developed based upon the modified strain gradient theory (MSGT) for the axial buckling analysis of functionally graded (FG) circular cylindrical microshells. It is assumed that the material properties of FG materials, which obey a simple power-law distribution, vary through the thickness direction. The principle of virtual work is utilized to formulate the governing equations and corresponding boundary conditions. Numerical results are presented for the axial buckling of FG circular cylindrical microshells subject to simply-supported end conditions and the effects of material length scale parameter, material property gradient index, length-to-radius ratio and circumferential mode number on the size-dependent critical buckling load are extensively studied. For comparison purpose, the critical buckling loads predicted by modified couple stress theory (MCST) and classical theory (CT) are also presented. Results show that the size effect plays an important role for lower values of dimensionless length scale parameter. Moreover, it is observed that the critical buckling loads obtained based on MSGT are greater than those obtained based on MCST and CT.     

Nonlinear vibration analysis of micro-plates based on strain gradient elasticity theory

In this study, non-linear free vibration of micro-plates based on strain gradient elasticity theory is investigated. A general form of Mindlin’s first-strain gradient elasticity theory is employed to obtain a general Kirchhoff micro-plate formulation. The von Karman strain tensor is used to capture the geometric non-linearity. The governing equations of motion and boundary conditions are obtained in a variational framework. The Homotopy analysis method is employed to obtain an accurate analytical expression for the non-linear natural frequency of vibration. For some specific values of the gradient-based material parameters, the general plate formulation can be reduced to those based on some special forms of strain gradient elasticity theory. Accordingly, three different micro-plate formulations are introduced, which are based on three special strain gradient elasticity theories. It is found that both geometric non-linearity and size effect increase the natural frequency of vibration. In a micro-plate having a thickness comparable with the material length scale parameter, the strain gradient effect on increasing the non-linear natural frequency is higher than that of the geometric non-linearity. By increasing the plate thickness, the strain gradient effect decreases or even diminishes. In this case, geometric non-linearity plays the main role on increasing the natural frequency of vibration. In addition, it is shown that for micro-plates with some specific thickness to length scale parameter ratios, both geometric non-linearity and size effect have significant role on increasing the frequency of non-linear vibration.     

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.     

Couple stress based strain gradient theory for elasticity

The deformation behavior of materials in the micron scale has been experimentally shown to be size dependent. In the absence of stretch and dilatation gradients, the size dependence can be explained using classical couple stress theory in which the full curvature tensor is used as deformation measures in addition to the conventional strain measures. In the couple stress theory formulation, only conventional equilibrium relations of forces and moments of forces are used. The couple's association with position is arbitrary. In this paper, an additional equilibrium relation is developed to govern the behavior of the couples. The relation constrained the couple stress tensor to be symmetric, and the symmetric curvature tensor became the only properly conjugated high order strain measures in the theory to have a real contribution to the total strain energy of the system. On the basis of this modification, a linear elastic model for isotropic materials is developed. The torsion of a cylindrical bar and the pure bending of a flat plate of infinite width are analyzed to illustrate the effect of the modification.     

Torsional vibration of functionally graded nano-rod under magnetic field supported by a generalized torsional foundation based on nonlocal elasticity theory

Abstract

This article contains the nonlocal elasticity theory to capture size effects in functionally graded (FG) nano-rod under magnetic field supported by a torsional foundation. Torque effect of an axial magnetic field on an FG nano-rod has been defined using Maxwell’s relation. The material properties were assumed to vary according to the power law in radial direction. The Navier equation and boundary conditions of the size-dependent FG nano-rod were derived by the Hamilton’s principle. These equations were solved by employing the generalized differential quadrature method (GDQM). Presented model has the ability to turn into the classical model if the material length scale parameter is taken to be zero. The effects of some parameters, such as inhomogeneity constant, magnetic field and small-scale parameter, were studied. As an important result of this study can be stated that an FG nano-rod model based on the nonlocal elasticity theory behaves softer and has smaller natural frequency.     

Linear and nonlinear torsional free vibration of functionally graded micro/nano-tubes based on modified couple stress theory

The linear and nonlinear torsional free vibration analyses of functionally graded micro/nano-tubes (FGMTs) are analytically investigated based on the couple stress theory. The employed non-classical continuum theory contains one material length scale parameter, which can capture the small scale effect. The FGMT model accounts for the through-radius power-law variation of a two-constituent material. Hamilton’s principle is used to develop the non-classical nonlinear governing equation. To study the effect of the boundary conditions, two types of end conditions, i.e., fixed-fixed and fixed-free, are considered. The derived boundary value governing equation is of the fourthorder, and is solved by the homotopy analysis method (HAM). This method is based on the Taylor series with an embedded parameter, and is capable of providing very good approximations by means of only a few terms, if the initial guess and the auxiliary linear operator are properly selected. The analytical expressions are developed for the linear and nonlinear natural frequencies, which can be conveniently used to investigate the effects of the dimensionless length scale parameter, the material gradient index, and the vibration amplitude on the natural frequencies of FGMTs.     

Free vibration analysis of functionally graded material beams based on Levinson beam theory

Free vibration response of functionally graded material (FGM) beams is studied based on the Levinson beam theory (LBT). Equations of motion of an FGM beam are derived by directly integrating the stress-form equations of elasticity along the beam depth with the inertial resultant forces related to the included coupling and higherorder shear strain. Assuming harmonic response, governing equations of the free vibration of the FGM beam are reduced to a standard system of second-order ordinary differential equations associated with boundary conditions in terms of shape functions related to axial and transverse displacements and the rotational angle. By a shooting method to solve the two-point boundary value problem of the three coupled ordinary differential equations, free vibration response of thick FGM beams is obtained numerically. Particularly, for a beam with simply supported edges, the natural frequency of an FGM Levinson beam is analytically derived in terms of the natural frequency of a corresponding homogenous Euler-Bernoulli beam. As the material properties are assumed to vary through the depth according to the power-law functions, the numerical results of frequencies are presented to examine the effects of the material gradient parameter, the length-to-depth ratio, and the boundary conditions on the vibration response.     

A size-dependent Kirchhoff micro-plate model based on strain gradient elasticity theory

A size-dependent Kirchhoff micro-plate model is developed based on the strain gradient elasticity theory. The model contains three material length scale parameters, which may effectively capture the size effect. The model can also degenerate into the modified couple stress plate model or the classical plate model, if two or all of the material length scale parameters are taken to be zero. The static bending, instability and free vibration problems of a rectangular micro-plate with all edges simple supported are carried out to illustrate the applicability of the present size-dependent model. The results are compared with the reduced models. The present model can predict prominent size-dependent normalized stiffness, buckling load, and natural frequency with the reduction of structural size, especially when the plate thickness is on the same order of the material length scale parameter.     

Non-uniform rational B-spline based free vibration analysis of axially functionally graded tapered Timoshenko curved beams

The free vibration of axially functionally graded (FG) tapered Timoshenko curved beams is studied with the numerical approach. By using the non-uniform rational B-spline (NURBS) basis functions, the exact geometry and the generalized displacement field are formulated. Variable geometric parameters and material properties, including the curvature, cross-sectional area, area moment of inertia, mass density, and Young’s modulus, are expanded as functions of the coordinate in a parametric domain. Based on Hamilton’s principle, the weak formulation is derived by applying a refined constitutive relation which considers the thickness effect. Natural frequencies and mode shapes are obtained from the eigenvalue equation. Circular, elliptic, and parabolic curved beams are considered in numerical examples. The obtained results are in good agreement with those in the existing studies and those calculated by the finite element software ANSYS. Moreover, the effects of the material gradient, taper ratio, slenderness ratio, and heightspan ratio on vibration behaviors are discussed.     

A micro scale Timoshenko beam model based on strain gradient elasticity theory

A micro scale Timoshenko beam model is developed based on strain gradient elasticity theory. Governing equations, initial conditions and boundary conditions are derived simultaneously by using Hamilton's principle. The new model incorporated with Poisson effect contains three material length scale parameters and can consequently capture the size effect. This model can degenerate into the modified couple stress Timoshenko beam model or even the classical Timoshenko beam model if two or all material length scale parameters are taken to be zero respectively. In addition, the newly developed model recovers the micro scale Bernoulli–Euler beam model when shear deformation is ignored. To illustrate the new model, the static bending and free vibration problems of a simply supported micro scale Timoshenko beam are solved respectively. Numerical results reveal that the differences in the deflection, rotation and natural frequency predicted by the present model and the other two reduced Timoshenko models are large as the beam thickness is comparable to the material length scale parameter. These differences, however, are decreasing or even diminishing with the increase of the beam thickness. In addition, Poisson effect on the beam deflection, rotation and natural frequency possesses an interesting “extreme point” phenomenon, which is quite different from that predicted by the classical Timoshenko beam model.     

Wave propagation responses of porous bi-directional functionally graded magneto-electro-elastic nanoshells via nonlocal strain gradient theory

This study examines the wave propagation characteristics for a bi-directional functional grading of barium titanate(BaTiO₃) and cobalt ferrite(CoFe₂O₄) porous nanoshells,the porosity distribution of which is simulated by the honeycomb-shaped symmetrical and asymmetrical distribution functions.The nonlocal strain gradient theory(NSGT) and first-order shear deformation theory are used to determine the size effect and shear deformation,respectively.Nonlocal governin...     

Experiments and theory in strain gradient elasticity

Conventional strain-based mechanics theory does not account for contributions from strain gradients. Failure to include strain gradient contributions can lead to underestimates of stresses and size-dependent behaviors in small-scale structures. In this paper, a new set of higher-order metrics is developed to characterize strain gradient behaviors. This set enables the application of the higher-order equilibrium conditions to strain gradient elasticity theory and reduces the number of independent elastic length scale parameters from five to three. On the basis of this new strain gradient theory, a strain gradient elastic bending theory for plane-strain beams is developed. Solutions for cantilever bending with a moment and line force applied at the free end are constructed based on the new higher-order bending theory. In classical bending theory, the normalized bending rigidity is independent of the length and thickness of the beam. In the solutions developed from the higher-order bending theory, the normalized higher-order bending rigidity has a new dependence on the thickness of the beam and on a higher-order bending parameter, b_h. To determine the significance of the size dependence, we fabricated micron-sized beams and conducted bending tests using a nanoindenter. We found that the normalized beam rigidity exhibited an inverse squared dependence on the beam's thickness as predicted by the strain gradient elastic bending theory, and that the higher-order bending parameter, b_h, is on the micron-scale. Potential errors from the experiments, model and fabrication were estimated and determined to be small relative to the observed increase in beam's bending rigidity. The present results indicate that the elastic strain gradient effect is significant in elastic deformation of small-scale structures.     

A nonlocal strain gradient shell model incorporating surface effects for vibration analysis of functionally graded cylindrical nanoshells

In this paper, a novel size-dependent functionally graded(FG) cylindrical shell model is developed based on the nonlocal strain gradient theory in conjunction with the Gurtin-Murdoch surface elasticity theory. The new model containing a nonlocal parameter, a material length scale parameter, and several surface elastic constants can capture three typical types of size effects simultaneously, which are the nonlocal stress effect, the strain gradient effect, and the surface energy effects. With the help of Hamilton's principle and first-order shear deformation theory, the non-classical governing equations and related boundary conditions are derived. By using the proposed model, the free vibration problem of FG cylindrical nanoshells with material properties varying continuously through the thickness according to a power-law distribution is analytically solved, and the closed-form solutions for natural frequencies under various boundary conditions are obtained. After verifying the reliability of the proposed model and analytical method by comparing the degenerated results with those available in the literature, the influences of nonlocal parameter, material length scale parameter, power-law index, radius-to-thickness ratio, length-to-radius ratio, and surface effects on the vibration characteristic of functionally graded cylindrical nanoshells are examined in detail.     

A nonlinear microbeam model based on strain gradient elasticity theory with surface energy

A microscale nonlinear Bernoulli–Euler beam model on the basis of strain gradient elasticity with surface energy is presented. The von Karman strain tensor is used to capture the effect of geometric nonlinearity. Governing equations of motion and boundary conditions are obtained using Hamilton’s principle. In particular, the developed beam model is applicable for the nonlinear vibration analysis of microbeams. By employing a global Galerkin procedure, the ordinary differential equation corresponding to the first mode of nonlinear vibration for a simply supported microbeam is obtained. Numerical investigations show that in a microbeam having a thickness comparable with its material length scale parameter, the strain gradient effect on increasing the beam natural frequency is higher than that of the geometric nonlinearity. By increasing the beam thickness, the strain gradient effect decreases or even diminishes. In this case, geometric nonlinearity plays the main role on increasing the natural frequency of vibration. In addition, it is shown that for beams with some specific thickness-to-length parameter ratios, both geometric nonlinearity and size effect have significant role on increasing the frequency of nonlinear vibration.     

热环境下功能梯度夹层双曲壳三维自由振动分析

功能梯度夹层双曲壳结构广泛应用在航空航天、海洋工程等领域中,对于该类结构的动力学特性研究非常重要。本文以热环境下功能梯度夹层双曲壳为研究对象,在三阶剪切变形理论的基础上,考虑横向拉伸作用的影响提出了一种新的位移场,假设材料的物性参数与温度有关,且沿厚度方向表示为幂律函数。利用Hamilton原理得到简支边界条件下功能梯度夹层双曲壳三维振动系统动力学方程,利用Navier法求得两种不同夹层类型的系统固有频率。研究了几何物理参数和温度场对功能梯度夹层双曲壳自由振动固有频率的影响。     

Analysis of micro-sized beams for various boundary conditions based on the strain gradient elasticity theory

Bending analysis of micro-sized beams based on the Bernoulli-Euler beam theory is presented within the modified strain gradient elasticity and modified couple stress theories. The governing equations and the related boundary conditions are derived from the variational principles. These equations are solved analytically for deflection, bending, and rotation responses of micro-sized beams. Propped cantilever, both ends clamped, both ends simply supported, and cantilever cases are taken into consideration as boundary conditions. The influence of size effect and additional material parameters on the static response of micro-sized beams in bending is examined. The effect of Poisson’s ratio is also investigated in detail. It is concluded from the results that the bending values obtained by these higher-order elasticity theories have a significant difference with those calculated by the classical elasticity theory.     

Size-dependent free vibration analysis of composite laminated Timoshenko beam based on new modified couple stress theory

A micro-scale free vibration analysis of composite laminated Timoshenko beam (CLTB) model is developed based on the new modified couple stress theory. In this theory, a new anisotropic constitutive relation is defined for modeling the CLTB. This theory uses rotation–displacement as dependent variable and contains only one material length scale parameter. Hamilton’s principle is employed to derive the governing equations of motion and boundary conditions. This new model can be reduced to composite laminated Bernoulli–Euler beam model of the couple stress theory. An example analysis of free vibration of the cross-ply simply supported CLTB model is adopted, and an explicit expression of analysis solution is given. Additionally, the numerical results show that the present beam models can capture the scale effects of the natural frequencies of the micro-structure.     

考虑尺度影响的平面正交各向异性功能梯度微梁的屈曲分析

基于新修正偶应力理论,建立了能描述尺度效应的各向异性功能梯度微梁的屈曲分析模型。基于最小势能原理推导了控制方程及边界条件,并以简支梁为例分析了屈曲载荷及尺度效应受材料尺度参数和几何尺寸的影响。算例结果表明,在材料几何尺寸较小时,本文模型预测到的屈曲载荷明显大于传统理论的结果,有效地反映了尺度效应。几何尺寸较大时,尺度效应消失,本文模型将自动退化为传统宏观模型。模型反映出不同方向上的尺度参数对各向异性材料影响的效果不同。     

An efficient shear deformation theory for vibration of functionally graded plates

This paper presents an efficient shear deformation theory for vibration of functionally graded plates. The theory accounts for parabolic distribution of the transverse shear strains and satisfies the zero traction boundary conditions on the surfaces of the plate without using shear correction factors. The mechanical properties of functionally graded plate are assumed to vary according to a power law distribution of the volume fraction of the constituents. Equations of motion are derived from the Hamilton??s principle. Analytical solutions of natural frequency are obtained for simply supported plates. The accuracy of the present solutions is verified by comparing the obtained results with those predicted by classical theory, first-order shear deformation theory, and higher-order shear deformation theory. It can be concluded that the present theory is not only accurate but also simple in predicting the natural frequencies of functionally graded plates.