Surface Stress Effects Research Articles

Postbuckling of Unknown-Length Nanobeam Considering the Effects of Nonlocal Elasticity and Surface Stress

The objective of this paper is to study the postbuckling behaviors of an unknown-length nanobeam combined with small-scale effects. The concept of variable-arc-length elastica is firstly applied on the problem of nanobeams. The span length is not changed while the arc length is varied increasingly. The nanobeam is on a clamped support at one end, while the other end is an overhanging part through a frictionless slot subjected to axial compression. At this end, the nanobeam is movable only in a horizontal direction. The governing equation is developed by the moment–curvature relationship based on the classical Euler–Bernoulli beam theory, including the effects of nonlocal elasticity, residual surface stress, and both combined effects. The shooting–optimization technique with two-point boundary condition is employed to solve the differential equations in this problem. The results, including nonlocal elasticity, reveal that nanobeams have decreased structural stiffness; meanwhile, the residual surface tension and both combined effects have increased strength. The postbuckling loads decrease as the arc length of nanobeams is increased. The equilibrium configurations are close to an anti-loop for very large deflections. The friction force at the nanoslot is also considered.

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.

On the postbuckling behavior of geometrically imperfect cylindrical nanoshells subjected to radial compression including surface stress effects

The main objective of the present study is to investigate the effect of surface stress on the nonlinear buckling and postbuckling behavior of cylindrical nanoshells with initial geometric imperfection subjected to radial compressive load. Gurtin–Murdoch elasticity theory is implemented into the classical shell theory to develop a size-dependent shell model which is capable to capture surface stress effects efficiently. In order to satisfy balance conditions on the surfaces of nanoshell, a linear variation through the thickness is considered for the normal stress component of the bulk. The principle of virtual work is put to use in order to formulate the non-classical governing differential equations. Afterwards, a boundary layer theory is employed including the nonlinear prebuckling deformations, initial geometric imperfection and large postbuckling deflections. Finally, a two-stepped singular perturbation methodology is utilized to obtain the size-dependent critical buckling loads and the postbuckling equilibrium paths of imperfect nanoshells corresponding to both lateral and hydrostatic pressure loading cases. It is found that for the positive and negative values of surface elastic constants, the both critical buckling load and critical end-shortening of nanoshell increase and decrease, respectively.

Surface effects on shape and topology optimization of nanostructures

We present a computational method for the optimization of nanostructures, where our specific interest is in capturing and elucidating surface stress and surface elastic effects on the optimal nanodesign. XFEM is used to solve the nanomechanical boundary value problem, which involves a discontinuity in the strain field and the presence of surface effects along the interface. The boundary of the nano-structure is implicitly represented by a level set function, which is considered as the design variable in the optimization process. Two objective functions, minimizing the total potential energy of a nanostructure subjected to a material volume constraint and minimizing the least square error compared to a target displacement, are chosen for the numerical examples. We present results of optimal topologies of a nanobeam subject to cantilever and fixed boundary conditions. The numerical examples demonstrate the importance of size and aspect ratio in determining how surface effects impact the optimized topology of nanobeams.

The Analysis of the Effect of Surface Stresses at Nanoscale

Based on classical elasticity theory, the effects of surface stresses on the nanosized contact problem in an elastic half-plane which contains a nanocylindrical hole are analyzed. Meanwhile, the effects of surface energy of the contact nanosized surface are considered. The complex variable function method is applied to derive the fundamental solution of the contact problem. As example, the deformation induced by a distributed traction of cosine function on the nanosized surface is analyzed in detail. The results tell some interesting characteristics in contact mechanics, which are different from those in classical elasticity theory.

Investigation of surface stress effect in 3D complex nano parts using FEM and modified boundary Cauchy–Born method

In this paper the surface effects in 3D complex parts are investigated via classical FEM and the modified boundary Cauchy–Born method, which are continuum based theories, and the results are compared with molecular dynamics simulations. The boundary Cauchy–Born covers the weakness of the surface Cauchy–Born in modeling edge and corner elements, so the results are in better agreement with molecular dynamics simulations. Using super convergent patch recovery method, MBCB shows more precise results in comparison with BCB as it can model acute and obtuse angles which cannot be modeled via BCB. The flexibility of MBCB in modeling 3D complex and non-right angle geometries are utilized to model industrial examples under free relaxation condition. Unlike samples in previous works, the complexity of geometry proposed here results in highly nonlinear problems where the MBCB method can no longer guarantee convergency of the solutions. Therefore, a new and modified arc-length method is developed and utilized to overcome this nonlinearity.

An exact solution for the nonlinear forced vibration of functionally graded nanobeams in thermal environment based on surface elasticity theory

In the present investigation, an exact solution is proposed for the nonlinear forced vibration analysis of nanobeams made of functionally graded materials (FGMs) subjected to thermal environment including the effect of surface stress. The material properties of functionally graded (FG) nanobeams vary through the thickness direction on the basis of a simple power law. The geometrically nonlinear beam model, taking into account the surface stress effect, is developed by implementing the Gurtin–Murdoch elasticity theory together with the classical Euler–Bernoulli beam theory and using a variational approach. Hamilton’s principle is utilized to obtain the nonlinear governing partial differential equation and corresponding boundary conditions. After that, the Galerkin technique is employed in order to convert the nonlinear partial differential equation into a set of nonlinear ordinary differential equations. This new set is then solved analytically based on the method of multiple scales which results in the frequency–response curves of FG nanobeams in the presence of surface stress effect. It is revealed that by increasing the beam thickness, the surface stress effect diminishes and the maximum amplitude of the stable response is shifted to the higher excitation frequencies.

Dynamic Response of Viscoelastic CNT Conveying Pulsating Fluid Considering Surface Stress and Magnetic Field

A viscoelastic carbon nanotube (CNT) conveying pulsating fluid is presented which is based on Euler–Bernoulli beam theory. Runge–Kutta scheme is chosen in order to illustrate the transverse and longitudinal behavior of structure. The effects of surface stress, magnetic field and nonlocal small-scale theory on motion of structure are expressed in this study. Equilibrium equations of CNT conveying pulsating fluid are obtained using energy method. Galerkin, differential quadrature and Runge–Kutta methods are applied to solve equations of motion. In this paper, the effect of pulsating fluid on longitudinal behavior of CNT and transverse displacement of CNT are presented. The static and dynamic transverse distributed loads and their effects on CNT are expressed. In this study, regions of CNT with chaotic, quasi-periodic and periodic behaviors are presented. Also the effects of various parameters such as distributed loads, surface stress and magnetic field on those regions are demonstrated. The results of this work could be helpful in design and manufacturing of nano-/micromechanical system in advanced medical applications such as drug delivery systems with magnetic field as a parametric controller.

Mathematical modeling of structural and near-surface non-homogeneities in thermoelastic thin films

We propose a model that is aimed to study the stress–strain state of nanoelements including size effects of surface stresses, effective elastic moduli and strength within the one-continuum approach. The size effects are caused by the non-homogeneity of the density. Non-homogeneity in the density distribution originates from taking into account the structure of the material and geometric irregularity of the real solid surface. Thermoelastic solid is considered as an open thermodynamic system. It is assumed that the material structure and real solid surface suddenly originate at the initial time while the phenomenological coefficients in the state equation for stress tensor depend on density. The obtained key system of equations is used to study the equilibrium state of thin film as a layer with the thickness comparable with characteristic sizes of near-surface and structural non-homogeneity regions. The multiscale type of size effects is indicated and the limits of applicability of the linear approximation are discussed.

Analysis of nanoindentation of functionally graded layered bodies with surface elasticity

Due to the high surface to volume ratio of the nanoscale domain, the surface stress effect is a major concern in the analysis of mechanical response of the nanomaterials and nanostructures. This paper presents a nonlinear variational numerical model for the analysis of frictionless indentation of a functionally graded layered homogeneous elastic bodies considering surface/interface stress effect in the context of contact mechanics. To accommodate the influence of surface/interface energy, the complete constitutive relation of the Gurtin–Murdoch elasticity theory is adopted, while the bulk material follows the classical theory of linear elasticity. The graded, nonhomogeneous property of the nanolayer follows a power law variation of the shear modulus, while a constant Poisson׳s ratio is considered. Utilizing the generalized isoparametric formulation, the graded finite element method is utilized to incorporate the material property gradient at the element scale. Throughout the indentation interface, the Lagrange multiplier approach is adopted to exactly satisfy the inequality contact constraints, where the indentation forces and the displacement field are treated as independent variables. Three nanoindentation problems of different natures are analyzed using the developed variational-based finite element computational model. The achieved results show the significance of the surface/interface energy, size-dependency, and the material distribution of functionally graded layer on the mechanical response of indentation processes.

Nonlocal plate model for dynamic pull-in instability analysis of circular higher-order shear deformable nanoplates including surface stress effect

In the present paper, dynamic pull-in instability and free vibration characteristics of circular higher-order shear deformable nanoplates subjected to hydrostatic and electrostatic forces are studied including surface stress effect. For this purpose, Eringen’s nonlocal elasticity continuum in conjunction with the Gurtin-Murdoch elasticity theory is incorporated into the classical higher-order shear deformation plate theory to develop size-dependent plate model able to consider both of small scale and surface stress effects. The non-classical governing differential equations are then discretized along with simply supported and clamped edge supports by employing generalized differential quadrature (GDQ) method. To evaluate the size-dependent pull-in voltage of nanoplates, the hydrostatic-electrostatic actuation is assumed to be calculated by neglecting the fringing field effects and utilizing the parallel plate approximation. It is demonstrated that the pull-in instability occurs at lower voltages for nanoplates with higher values of nonlocal parameters. Moreover, it is found that surface stress effect can increase or decrease the pull-in voltage of nanoplates which depends on the sign of surface elastic constants.

Vibration and buckling characteristics of functionally graded nanoplates subjected to thermal loading based on surface elasticity theory

The buckling and vibration responses of nanoplates made of functionally graded materials (FGMs) subjected to thermal loading are studied in prebuckling domain with considering the effect of surface stress. To accomplish this purpose, Gurtin–Murdoch elasticity theory is incorporated into the classical plate theory to develop a non-classical plate model including the surface effects. The material properties of FGM nanoplate are considered to be graded in the thickness direction on the basis of the power law function. Hamilton׳s principle is utilized to derive size-dependent governing differential equations of motion and associated boundary conditions. Selected numerical results are presented to indicate the importance of surface stress effect. It is revealed that in the presence of surface stress effect, the influence of material property gradient index on the critical thermal buckling load is more prominent for FGM nanoplates with lower length-to-thickness ratios. Also, by increasing the natural frequency of FGM nanoplate, the role of surface stress effect in the value of critical thermal buckling load is more prominent.

Nonlinear Vibrational Analysis of Nanobeams Embedded in an Elastic Medium including Surface Stress Effects

Due to size-dependent behavior of nanostructures, the classical continuum models are not applicable for the analyses at this submicron size. Surface stress effect is one of the most important matters which make the nanoscale structures have different properties compared to the conventional structures due to high surface to volume ratio. In the present study, nonlinear free vibrational characteristics of embedded nanobeams are investigated including surface stress effects. To this end, a thin surface layer is assumed on the upper and lower surfaces of the cross section to separate the surface and bulk of nanobeams with their own different material properties. Based on harmonic balance method, closed-form analytical solution is conducted for nonlinear vibrations to obtain natural frequencies of embedded nanobeams with and without considerations of surface elasticity and residual surface tension effects corresponding to the various values of nondimensional amplitude, elastic foundation modulus, and geometrical variables of the system. Selected numerical results are given to indicate the influence of each one in detail.

Surface tension and the mechanics of liquid inclusions in compliant solids.

Eshelby's theory of inclusions has wide-reaching implications across the mechanics of materials and structures including the theories of composites, fracture, and plasticity. However, it does not include the effects of surface stress, which has recently been shown to control many processes in soft materials such as gels, elastomers and biological tissue. To extend Eshelby's theory of inclusions to soft materials, we consider liquid inclusions within an isotropic, compressible, linear-elastic solid. We solve for the displacement and stress fields around individual stretched inclusions, accounting for the bulk elasticity of the solid and the surface tension (i.e. isotropic strain-independent surface stress) of the solid-liquid interface. Surface tension significantly alters the inclusion's shape and stiffness as well as its near- and far-field stress fields. These phenomena depend strongly on the ratio of the inclusion radius, R, to an elastocapillary length, L. Surface tension is significant whenever inclusions are smaller than 100L. While Eshelby theory predicts that liquid inclusions generically reduce the stiffness of an elastic solid, our results show that liquid inclusions can actually stiffen a solid when R<3L/2. Intriguingly, surface tension cloaks the far-field signature of liquid inclusions when R=3L/2. These results are have far-reaching applications from measuring local stresses in biological tissue, to determining the failure strength of soft composites.

Effects of pre-stress and surface stress on phonon thermal conductivity of rectangular Si nanowires

This work investigates theoretically the phonon property and thermal conductivity of rectangular silicon nanowires under pre-stress and surface stress. In the framework of elasticity theory, the effects of spatial confinement are considered in the phonon dispersion relation of a stressed nanowire. The surface energy, which brings about the variation of the elastic modulus of nanowire and the influence on the phonon property, is then involved. Under a pre-stress field, the acoustoelastic effect gives rise to the change of phonon properties and thermal conductivity. Our numerical results demonstrate that the applied surface stress and pre-stress field can alter the phonon dispersion relation of a silicon nanowire significantly. The phonon energy increases if the surface stress is negative and the pre-stress is positive, and vice versa. The changes of phonon dispersion relation as well as the various phonon scattering rates lead to the variation of phonon thermal conductivity, which is the consequence of the surface stress and pre-stress fields. We further elaborate the size and temperature dependence of phonon thermal conductivity under different applied surface stresses and pre-stress fields and suggest using the strain engineering to tune the thermal performance of semiconductor nanostructures.

Buckling Analysis of a Nanowire Lying on Winkler–Pasternak Elastic Foundation

This article studies the axial buckling of a nanowire (NW) lying on Winkler–Pasternak substrate medium with the Timoshenko beam theory. The surface effect of the NW is accounted for with the Steigmann–Ogden model. An explicit solution of the critical buckling force and its associated buckling mode are obtained analytically. The influences of the surface stress effect, the geometry of the NW, and the elastic foundation moduli on the buckling behavior are fully discussed.

Pulsating fluid induced dynamic instability of visco-double-walled carbon nano-tubes based on sinusoidal strain gradient theory using DQM and Bolotin method

This research deals with the dynamic instability analysis of double-walled carbon nanotubes (DWCNTs) conveying pulsating fluid under 2D magnetic fields based on the sinusoidal shear deformation beam theory (SSDBT). In order to present a realistic model, the material properties of DWCNTs are assumed viscoelastic using Kelvin–Voigt model. Considering the strain gradient theory for small scale effects, a new formulation of the SSDBT is developed through the Gurtin–Murdoch elasticity theory in which the effects of surface stress are incorporated. The surrounding elastic medium is described by a visco-Pasternak foundation model, which accounts for normal, transverse shear and damping loads. The van der Waals interactions between the adjacent walls of the nanotubes are taken into account. The size dependent motion equations and corresponding boundary conditions are derived based on the Hamilton’s principle. The differential quadrature method in conjunction with Bolotin method is applied for obtaining the dynamic instability region. The detailed parametric study is conducted, focusing on the combined effects of the nonlocal parameter, magnetic field, visco-Pasternak foundation, Knudsen number, surface stress and fluid velocity on the dynamic instability of DWCNTs. The results depict that the surface stress effects on the dynamic instability of visco-DWCNTs are very significant. Numerical results of the present study are compared with available exact solutions in the literature. The results presented in this paper would be helpful in design and manufacturing of nano/micro mechanical systems in advanced biomechanics applications with magnetic field as a parametric controller.

Nanometer-resolved mechanical properties around GaN crystal surface steps.

The mechanical properties of surfaces and nanostructures deviate from their bulk counterparts due to surface stress and reduced dimensionality. Experimental indentation-based techniques present the challenge of measuring these effects, while avoiding artifacts caused by the measurement technique itself. We performed a molecular dynamics study to investigate the mechanical properties of a GaN step of only a few lattice constants step height and scrutinized its applicability to indentation experiments using a finite element approach (FEM). We show that the breakdown of half-space symmetry leads to an “artificial” reduction of the elastic properties of comparable lateral dimensions which overlays the effect of surface stress. Contact resonance atomic force microscopy (CR-AFM) was used to compare the simulation results with experiments.

Size-dependent resonance and buckling behavior of nanoplates with high-order surface stress effects

This work presents a theoretical study of the resonance frequency and buckling load of nanoplates with high-order surface stress model. A classical thin plate theory based on Kirchhoff–Love assumption is implemented with surface effects. Circular and rectangular nanoplates with simply supported end conditions are exemplified. The size-dependent solutions are compared with the simplified solutions based on simple surface stress model, and also on the classical theory of elasticity. We aim to explore the scope of applicability so that the modified continuum mechanics model could serve as a refined approach in the prediction of mechanical behavior of nanoplates.

Elastic relaxation in 3D epitaxial nanoisland with strain-dependent surface stress effect

Surface stress effect plays an important role in determining the properties of nanostructures. In lattice-mismatched 3D epitaxial systems, elastic relaxation could effectively reduce the natural misfit between the nanoisland and substrate. This work proposes a systematic study on the elastic relaxation in 3D epitaxial nanoisland with strain-dependent surface stress effect. The analytical methodology is formulated for general anisotropic nanostructures with general geometries (size and shape). An example of a copper cuboid is provided to demonstrate how the finite size and surface property of the nanoisland would influence the elastic relaxation between the nanoisland and substrate. The influence of surface stiffness is shown to be more significant in the lateral dimension than the height dimension. More importantly, this work indicates that surface stiffness becomes important to determine the overall property of the nanoisland when its size is lower than 10nm, so surface stress should be considered to be strain-dependent in such situations. This would be important for the fabrication of high quality nanostructures.