Surface stress effects on the critical buckling strains of silicon nanowires

The objective of this paper is to quantify how nanoscale surface stresses impact the critical buckling strains of silicon nanowires. These insights are gained by using nonlinear finite element calculations based upon a multiscale, finite deformation constitutive model that incorporates nanoscale surface stress and surface elastic effects to study the buckling behavior of silicon nanowires that have cross sectional dimensions between 10 and 25 nm under axial compressive loading. The key finding is that, in contrast to existing surface elasticity solutions, the critical buckling strains are found to show little deviation from the classical bulk Euler solution. The present results suggest that accounting for axial strain relaxation due to surface stresses may be necessary to improve the accuracy and predictive capability of analytic linear surface elastic theories.     

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.     

Surface effects on the near-tip stress fields of a mode-II crack

On the physical nature, most crack tips are not ideally sharp but have a small curvature radius. Both surface energy and crack-root curvature affect the stress and displacement fields in the vicinity of the crack tip. In the present paper, a numerical method, which incorporates the effect of surface elasticity into the finite element method, is employed to study the surface effects on the mode-II crack tip fields. It is found that when the curvature radius of the crack root decreases to micro-/nanometers, surface elasticity has a significant influence on the stresses near the crack tip. For a mode-II crack, surface effects alter both the magnitude and position of the maximum stresses, as is different from a mode-I crack, in which case only the stress magnitude is influence by surface stresses.     

Nonlocal free vibration of orthotropic non-prismatic skew nanoplates

The free vibration of orthotropic non-prismatic skew nanoplate based on the first-order shear deformation theory (FSDT) in conjunction with Eringen’s nonlocal elasticity theory is presented. As a simple, accurate and low computational effort numerical method, the differential quadrature method (DQM) is employed to solve the related differential equations. For this purpose, after deriving the equations of motion and the related boundary conditions, they are transformed from skewed physical domain to rectangular computational domain of DQM and accordingly discretized. After validating the formulation and method of solution, the effects of nonlocal parameter in combination with geometrical parameters and boundary conditions on the natural frequencies of the orthotropic skew nanoplates are investigated.     

Surface and nonlocal effects on the nonlinear free vibration of non-uniform nanobeams

The surface and nonlocal effects on the nonlinear flexural free vibrations of elastically supported non-uniform cross section nanobeams are studied simultaneously. The formulations are derived based on both Euler–Bernoulli beam theory (EBT) and Timoshenko beam theory (TBT) independently using Hamilton’s principle in conjunction with Eringen’s nonlocal elasticity theory. Green’s strain tensor together with von Kármán assumptions are employed to model the geometrical nonlinearity. The differential quadrature method (DQM) as an efficient and accurate numerical tool in conjunction with a direct iterative method is adopted to obtain the nonlinear vibration frequencies of nanobeams subjected to different boundary conditions. After demonstrating the fast rate of convergence of the method, it is shown that the results are in excellent agreement with the previous studies in the limit cases. The influences of surface free energy, nonlocal parameter, length of non-uniform nanobeams, variation of nanobeam width and elastic medium parameters on the nonlinear free vibrations are investigated.     

Surface effects at the nanoscale significantly reduce the effects of stress concentrators

At nanoscales, the large surface over volume ratio is shown to be instrumental in eliminating or significantly reducing the adverse effects of nanoscale stress concentrators (NSCs) such as impurities, inclusions, pores, and cracks. Using molecular dynamics (MD) simulations, Cu crystals with and without NSCs are strained in tension and in shear, at two strain rates, one being an order of magnitude larger than the other. Cube-shape crystals with periodic boundary conditions show sensitivity to NSCs similar to macrosize samples where fracture mechanics works well. For such crystals, atomistic defects cluster near the loaded surfaces, the clustering being stimulated significantly by the NSCs. Crystals with non-periodic boundary conditions, however, show insensitivity to NSCs, for the sample sizes examined herein, i.e., cubes up to about 30 nm side length. Atomistic defects do not cluster near the loading surfaces but rather distribute over the entire sample. Even though the spatial distribution of atomistic defects depends on the presence of NSCs, the total number of such defects is found to be independent of the presence of NSCs for the cubic crystals. The reason for this is the presence of a “vast” amount of surfaces, for the non-periodic boundary conditions case, where numerous atomistic defects initiate, making the number of defects initiating from NSCs insignificant. Provided that the average energy in creating these defects is constant, a robust explanation of the insignificance of NSCs emerges.     

Small scale effect on the free vibration of orthotropic arbitrary straight-sided quadrilateral nanoplates

As a first endeavor, the free vibration of orthotropic arbitrary straight-sided quadrilateral nanoplates is investigated using the nonlocal elasticity theory. The formulation is derived based on the first order shear deformation theory (FSDT). The solution procedure is based on the transformation of the governing equations from physical domain to computational domain and then discretization of the spatial derivatives by employing the differential quadrature method (DQM) as an efficient and accurate numerical tool. The formulation and the method of the solution are firstly validated by carrying out the comparison studies for the isotropic and orthotropic rectangular plates against existing results in literature. Then, the effects of nonlocal parameter in combination with the geometrical shape parameters, thickness-to-length ratio and the boundary conditions on the frequency parameters of the nanoplates are investigated.     

Nonlinear vibration of the piezoelectric nanobeams based on the nonlocal theory

This paper investigates the nonlinear vibration of the piezoelectric nanobeams based on the nonlocal theory and Timoshenko beam theory. The piezoelectric nanobeam is subjected to an applied voltage and a uniform temperature change. The nonlinear governing equations and boundary conditions are derived by using the Hamilton principle and discretized by using the differential quadrature (DQ) method. A direct iterative method is employed to determine the nonlinear frequencies and mode shapes of the piezoelectric nanobeams. A detailed parametric study is conducted to study the influences of the nonlocal parameter, temperature change and external electric voltage on the size-dependent nonlinear vibration characteristics of the piezoelectric nanobeams.     

Nonlinear buckling and postbuckling behavior of cylindrical nanoshells subjected to combined axial and radial compressions incorporating surface stress effects

In the present study, the Gurtin-Murdoch elasticity theory, as a theory capable of capturing size effects, is implemented to predict the nonlinear buckling and postbuckling response of cylindrical nanoshells under combined axial and radial compressive loads in the presence of surface stress effects. For this purpose, a size-dependent shell mode containing geometric nonlinearity is proposed within the framework of the classical shell theory. Because it is necessary to satisfy balance conditions on the surfaces of nanoshell, it is assumed that the normal stress component of the bulk varies linearly through the shell thickness. On the basis of a variational formulation using the principle of virtual work, the non-classical governing differential equations are derived. Subsequently, a boundary layer theory is employed including the nonlinear prebuckling deformations and the large deflections in the postbuckling regime. Then a two-stepped perturbation methodology is utilized to obtain the size-dependent critical buckling loads and the postbuckling equilibrium paths of nanoshells corresponding to the axial dominated and radial dominated loading cases. It is revealed that in the radial dominated loading case, a positive value of surface elastic constants leads to increase the critical buckling load but decrease the critical end-shortening of nanoshell. However, in the axial dominated loading case, surface elastic constants with positive sign causes to increase the both critical buckling load and critical end-shortening of nanoshell.     

考虑尺度依赖的平面正交各向异性功能梯度微梁的自由振动分析

基于一种新修正偶应力理论建立了微尺度平面正交各向异性功能梯度梁的自由振动模型。模型中包含两个材料尺度参数,能够分别描述两个正交方向上不同程度的尺度效应。当梁的几何尺寸远大于材料尺度参数时,本文模型亦可自动退化为相应的传统宏观模型。基于哈密顿原理推导了运动控制方程并以简支梁的自由振动为例分析了几何尺寸、功能梯度变化指数等对尺度效应产生的影响。算例结果表明:采用本文模型所预测的梁自振频率总是大于传统理论的结果,即捕捉到了尺度效应。尺度效应会随着梁几何尺寸的增大而逐渐减弱并在几何尺寸远大于尺度参数时消失;高阶自振频率所体现出的尺度效应较低阶自振频率更加明显。此外,功能梯度变化指数对尺度效应也有一定的影响。     

Surface effects on mode-I crack tip fields: A numerical study

Surface energy often significantly influences the deformation and failure behavior of materials and devices at the nanoscale. However, how it alters the local deformation around a crack tip remains unclear. In the present paper, we investigate the surface effects on the near-tip fields of a mode-I blunt crack (or notch). The theory of surface elasticity is incorporated into the finite element method. It is found that when the curvature radius of the crack root shrinks to nanometers, surface effects considerably affect the local stress distributions near the crack tip. We also calculate the J-integral, which is almost independent of surface effects except when the integral path approaches the crack tip. This demonstrates that surface effects are localized in a small zone around the crack tip, where the classical fracture mechanics solutions neglecting surface effects should be modified.     

Buckling of composite thin walled beams by refined theory

This work presents the buckling analysis of laminated composite thin walled structures by the 1D finite element based unified higher-order models obtained within the framework of the Carrera Unified Formulation (CUF). In the present study, the refined beam theories are obtained on the basis of Taylor-type expansions. The finite element analysis has been chosen to easily handle arbitrary geometries as well as boundary conditions. Buckling behavior of laminated composite beam and flat panels are analyzed to illustrate the efficacy of the present formulation and various types of buckling modes are observed depending on the geometrical and material parameters. It is observed that the lower order models are unable to deal with torsion.     

Nonlocal and surface effects on the buckling behavior of flexoelectric sandwich nanobeams

In this article, an analytical approach is presented to study the surface and flexoelectric effects on the buckling characteristics of an embedded piezoelectric sandwich nanobeam. According to the nonlocal elasticity theory, the flexoelectricity is believed to be authentic for size-dependent properties in nanostructures. The boundary conditions and the governing equations are derived by Hamilton's principle and are solved by Navier method. The results obtained from the present work show that the nonlocal term has an important reduction on the critical load and also the flexoelectricity shows an increasing influence on the buckling loads of the sandwich nanobeam, especially at lower thicknesses.     

On free vibration of piezoelectric nanospheres with surface effect

Surface effect responsible for some size-dependent characteristics can become distinctly important for piezoelectric nanomaterials with inherent large surface-to-volume ratio. In this paper, we investigate the surface effect on the free vibration behavior of a spherically isotropic piezoelectric nanosphere. Instead of directly using the well-known Huang-Yu surface piezoelectricity theory (HY theory), another general framework based on a thin shell layer model is proposed. A novel approach is developed to establish the surface piezoelectricity theory or the effective boundary conditions for piezoelectric nanospheres employing the state-space formalism. Three different sources of surface effect can be identified in the first-order surface piezoelectricity, i.e. the electroelastic effect, the inertia effect, and the thickness effect. It is found that the proposed theory becomes identical to the HY theory for a spherical material boundary if the transverse stress components are discarded and the electromechanical properties are properly defined. The nonaxisymmetric free vibration of a piezoelectric nanosphere with surface effect is then studied and an exact solution is obtained. In order to investigate the surface effect on the natural frequencies of piezoelectric nanospheres, numerical calculations are finally performed. Our numerical findings demonstrate that the surface effect, especially the thickness effect, may have a particularly significant influence on the free vibration of piezoelectric nanospheres. This work provides a more accurate prediction of the dynamic characteristics of piezoelectric nanospherical devices in nano-electro-mechanical systems.     

Surface effects on the bending,buckling and free vibration analysis of magneto-electro-elastic beams

Surface effects are responsible for the size dependence and should be taken into account for dielectric structures at nanoscale dimensions. By incorporating the effects of surface stress, surface piezoelectricity, surface elasticity and surface piezomagneticity, this paper investigates the bending, buckling and free vibration of magneto-electro-elastic (MEE) beams based on the Euler–Bernoulli beam theory. The governing differential equation and its corresponding boundary conditions are derived by Hamilton’s principle. The analytical solutions for the magneto-electro-elastic bending deflection, buckling magnetic potentials and frequency equations of MEE beams are obtained. In contrast to the previously published works, the positive surface stress is found to stiffen the MEE beams, as evidenced by the decrease in the deflections, the increase in the buckling magnet potentials and the increase in the resonant frequencies. Numerical studies show the importance of the surface effects, the electric and magnetic potentials and boundary conditions on the static and dynamic behavior of MEE beams. This work may be of special interest in the design and application of smart composite MEE beams.     

Surface effects on free vibration of piezoelectric functionally graded nanobeams using nonlocal elasticity

Free vibration of functionally graded material (FGM) nanobeams is investigated by considering surface effects including surface elasticity, surface stress, and surface density as well as the piezoelectric field using nonlocal elasticity theory. The balance conditions between the nanobeam bulk and its surfaces are satisfied assuming a cubic variation for the normal stress, ${\sigma_{zz}}$ , through the piezoelectric FG nanobeam thickness. Accordingly, the surface density is introduced into the governing equation of the free vibration of nanobeams. The results are obtained for various gradient indices, voltage values of the piezoelectric field, nanobeam lengths, and mode numbers. It is shown that making changes to voltage values and modifying mechanical properties of piezoelectric FGM nanobeams are two main approaches to achieve desired natural frequencies.     

An analytical study on the nonlinear free vibration of functionally graded nanobeams incorporating surface effects

Nonlinear free vibration of simply supported FG nanoscale beams with considering surface effects (surface elasticity, tension and density) and balance condition between the FG nanobeam bulk and its surfaces is investigated in this paper. The non-classical beam model is developed within the framework of Euler–Bernoulli beam theory including the von Kármán geometric nonlinearity. The component of the bulk stress, σ_zz, is assumed to vary cubically through the nanobeam thickness and satisfies the balance conditions between the FG nanobeam bulk and its surfaces. Accordingly, surface density is introduced into the governing equation of the nonlinear free vibration of FG nanobeams. The multiple scales method is employed as an analytical solution for the nonlinear governing equation to obtain the nonlinear natural frequencies of FG nanbeams. Several comparison studies are carried out to demonstrate the effect of considering the balance conditions on free nonlinear vibration of FG nanobeams. Lastly, the influences of the FG nanobeam length, volume fraction index, amplitude ratio, mode number and thickness ratio on the normalized nonlinear natural frequencies of the FG nanobeams are discussed in detail.     

Effects of size and surface on the elasticity of silicon nanoplates: Molecular dynamics and semi-continuum approaches

In this paper, the size effects on the elastic behavior of single crystal silicon nanoplates terminated by {100} surfaces is studied by means of molecular dynamics (MD) using a modified embedded atom method. The results indicate that the {100} surfaces undergo 2 × 1-type reconstruction, which significantly influences the mechanical properties of nanoplates. The simulations are carried out at room temperature and structural relaxation is performed. The effective Young's modulus, in extensional mode, is determined for different thicknesses. The surface energy, surface stress and surface elasticity of layers near the surfaces (non-bulk layers) are obtained. These surface properties are used as inputs for a recently developed two-dimensional plane-stress semi-continuum framework. The framework can be seen as the link between the surface effects calculated by atomistic simulations and the overall elastic behavior. The surface properties of nanoplates of a few layers are shown to deviate from thicker plates, suggesting a size dependence of surface parameters and, especially, surface energy. For thicknesses below 3 nm, there is a difference between the effective Young's modulus, calculated by the semi-continuum approach and that calculated directly by MD. The difference is due to the size dependence of surface parameters.     

Bending,buckling and vibration of size-dependent functionally graded annular microplates

The bending, buckling and free vibration of annular microplates made of functionally graded materials (FGMs) are investigated in this paper based on the modified couple stress theory and Mindlin plate theory. This microplate model incorporates the material length scale parameter that can capture the size effect in FGMs. The material properties of the FGM microplates are assumed to vary in the thickness direction and are estimated through the Mori–Tanaka homogenization technique. The higher-order governing equations and boundary conditions are derived by using Hamilton’s principle. The differential quadrature (DQ) method is employed to discretize the governing equations and to determine the deflection, critical buckling load and natural frequencies of FGM microplates. A parametric study is then conducted to investigate the influences of the length scale parameter, gradient index and inner-to-outer radius ratio on the bending, buckling and vibration characteristics of FGM microplates with hinged–hinged and clamped–clamped supports. The results show that the size effect on the bending, buckling and vibration characteristics is significant when the ratio of the microplate thickness to the material length scale parameter is smaller than 10.     

基于新修正偶应力理论的斜交铺设层合Kirchhoff板模型与尺度效应

基于新修正偶应力理论, 建立了只含有1个尺度参数的复合材料斜交铺设层合Kirchhoff板模型, 并分析了铺设角对尺度效应的影响。采用虚功原理推导了任意铺设角的层合Kirchhoff板的弯曲方程和边界条件。通过新模型给出了四边简支反对称角铺设微尺度层合Kirchhoff板的解析解。结果表明:细观尺度各向异性层合板的尺度效应并不仅仅受其几何尺寸的影响, 还受铺设角的影响;铺设角对尺度效应可能产生非常显著的影响。