Surface Stress Effects Research Articles

Effect of surface stress on the stiffness of micro/nanocantilevers: Nanowire elastic modulus measured by nano-scale tensile and vibrational techniques

Surface stress induced stiffness change of micro/nanocantilevers is reviewed and rigorously examined in this work. The self-equilibrium strain field of micro/nanocantilevers carrying an inherent surface stress on substrate is derived by resorting to the generalized Young-Laplace equation. It is found that the mechanism responsible for the observed stiffness change of micro/nano cantilevers originating from surface stress cannot be attributed to the development of in-plane stress near the clamp. Based on the analysis, two loading modes used in the mechanical test experiments performed on nanowire (NW) are theoretically investigated in detail: tension and electrically-induced-vibration. Lattice distortions arising from surface stress, coupled with that induced by residual strain, are shown to play a significant role in the elastic modulus measurement of NWs using an electric-field-induced vibrational mode, but have no influences on the tensile testing mode. The analytical results are validated by comparisons with molecular dynamic simulations and experimental measurements. The present results are useful in interpreting differences in observed size-dependent elasticity of NWs and developing the nano- and micro-mechanical testing techniques.

Tailoring diffusion-induced stresses of core-shell nanotube electrodes in lithium-ion batteries

Carbon-coated electrode nanoparticles enhance the cycling stability of lithium-ion batteries due to their intrinsic electric conductivity and excellent tolerance to mechanical stress. To study diffusion-induced stresses of these nanocomposites, nanotube electrodes wrapped with carbon shells are investigated including the effects of surface stress. The results of our model show that diffusion-induced stresses strongly depend on the thickness of carbon layer, which should be tuned to endure material strengths, avoiding mechanical fracture. In addition, surface tension produces compressive stresses through the electrode materials, even a tensile state can turn into a state of compressive stress, which may become a resistance to brittle fracture.

Effect of Surface Stress on Stress Intensity Factors of a Nanoscale Crack via Double Cantilever Beam Model

This paper studies the influence of surface elasticity on crack growth for a nanoscale crack advance. A crack is modeled as a double cantilever beam with consideration of surface stress. Using the Euler-Bernoulli beam theory incorporating with surface effects, a governing equation of static bending is derived and bending solution of a cantilever nanowire is obtained for a concentrated force at the free end. Based on the viewpoint of energy balance, the elastic strain energy is given and energy release rate is determined. The influences of the Surface stress and the surface elasticity on crack growth are discussed. Obtained results indicate that consideration of the surface effects decreases stress intensity factors or energy release rates. The residual surface tension impedes propagation of a nanoscale crack and apparent fracture toughness of nanoscale materials is effectively enhanced.

Effects of Free Surface and Heterogeneous Residual Internal Stress on Stress‐Driven Grain Growth in Nanocrystalline Metals

By reevaluating the experimental study of Zhang et al. (2005), here we demonstrate that the extent of grain growth, previously proposed to be solely driven by external stress, may have been significantly overestimated. A new physical mechanism, termed as free surface assisted stress‐driven grain growth (or self‐mechanical annealing), is proposed and discussed in detail. Representing the cooperative effect of free surface and heterogeneous residual internal stress, the proposed mechanism is considered more favorable than the traditional pure stress‐driven mechanism for interpreting the abnormal grain growth widely observed in deforming nanocrystalline metals at room temperature.

A sixth-order compact finite difference method for non-classical vibration analysis of nanobeams including surface stress effects

A non-classical model for the free vibrations of nanobeams accounting for surface stress effects is developed in this study. Based on Gurtin–Murdoch elasticity theory, the influence of surface stress is incorporated into the Euler–Bernoulli beam theory. A compact finite difference method (CFDM) of sixth order is employed for discretizing the non-classical governing differential equation to obtain the natural frequencies of nanobeams subject to different boundary conditions. To check the validity of the present numerical solution, based on an exact solution, an explicit formula for the fundamental frequency of simply-supported nanobeams is obtained. Good agreement between the results of exact and numerical solutions is achieved, confirming the validity and accuracy of the present numerical scheme. The comparison between the results generated by CFDM with those obtained by the conventional finite difference method (FDM) further reveals the advantages of the compact method over its classical counterpart. The influences of beam thickness, surface density, surface residual stress, surface elastic constants, and boundary conditions on the natural frequencies of nanobeams are also investigated. It is indicated that the effect of surface stress on the vibrational response of a nanobeam is dependent on its aspect ratio and thickness.

Diffusion-induced stresses of electrode nanomaterials in lithium-ion battery: The effects of surface stress

Nanomaterials offer large reaction surfaces making for high-rate lithium-ion transfer and fewer constraints to avoid fracture. Nevertheless, surface effect arises inevitably due to so high surface-to-volume ratio. Accordingly, the fundamental framework of surface stress is involved to study diffusion-induced stresses within electrode nanoparticles in this work. As simple one-dimension models, solid and hollow nanowire electrode particles are investigated. The results show that surface tensile stress produces compressive stresses through the electrode materials, especially reducing maximum tensile stress, which may become a resistance to brittle fracture. Owing to high special surface area, it is demonstrated that diffusion-induced stresses for hollow materials are largely reduced compared to solid electrode materials. The influences of surface modulus on diffusion-induced stresses are much stronger under generalized plane strain condition in comparison with plane strain condition. Analysis based on the Tresca criterion indicates that shear failure may occur at the inner surface with decreasing radius.

Correlation between Surface Stress and Apparent Young’s Modulus of Top-Down Silicon Nanowires

In this work, we report experimental evidence of surface stress effects on the mechanical properties of silicon nanostructures. As-fabricated, top-down silicon nanowires (SiNWs) are bent up without any applied force. This self-buckling is related to the surface relaxation that reaches an equilibrium with bulk deformation due to the material elasticity. We measure the SiNW self-deformation by atomic force microscopy (AFM), and we apply a simple physical model in order to give an estimation of the surface stress. If the equilibrium is altered by a nanoforce, applied by an AFM tip, nanowires find a new equilibrium condition bending down (mechanical bistability). In this work, for the first time, we report a clear and quantitative relationship between the SiNWs' apparent Young's modulus, measured by force-deflection spectroscopy, and the estimated value of surface stress, obtained by self-buckling measurements taking into account the Young's modulus of bulk silicon. This is an experimental confirmation that the surface stress is fundamental in determining mechanical properties of SiNWs, and that the elastic behavior of nanostructures strongly depends on their surfaces.

Surface Stress Effect on the Pull-In Instability of Hydrostatically and Electrostatically Actuated Rectangular Nanoplates With Various Edge Supports

This paper is aimed to investigate the size-dependent pull-in behavior of hydrostatically and electrostatically actuated rectangular nanoplates including surface stress effects based on a modified continuum model. To this end, based on the Gurtin–Murdoch theory and Hamilton's principle, the governing equation and corresponding boundary conditions of an actuated nanoplate are derived; the step-by-step linearization scheme and the differential quadrature (GDQ) method are used to discretize the governing equation and associated boundary conditions. The effects of the thickness of the nanoplate, surface elastic modulus and residual surface stress on the pull-in instability of the nanoplate are investigated. Plates made of two different materials including aluminum (Al) and silicon (Si) are selected to explain the variation of the pull-in voltage and pressure with respect to plate thickness.

Bending and buckling of nanowires including the effects of surface stress and nonlocal elasticity

Abstract This paper presents the bending deformation due to uniformly distributed load and buckling load of nanowires with various boundary conditions including surface stress and nonlocal elasticity effects. The analytical solutions for static displacement and buckling load of nanowires are derived and compared with those of numerical results obtained by the finite element method. The results indicate that the influences of surface stress and nonlocal elasticity affects directly both the displacement and buckling load. The effect of surface stress is to increase the stiffness of nanowires. Therefore, the nanowires for all boundary conditions show stiffer behavior in comparison with classical Euler beam. The effect of nonlocal elasticity can result in either stiffer or softer behaviors of nanowires depending on the boundary conditions. In case of buckling loads, the surface stress increases the buckling load significantly while the nonlocal elasticity is affected slightly. For nanowires including both effects, the results show that the buckling load is between one of the nanowires with surface stress and the one with nonlocal elasticity.

On the importance of surface elastic contributions to the flexural rigidity of nanowires

We present a theoretical model to calculate the flexural rigidity of nanowires from three-dimensional elasticity theory that incorporates the effects of surface stress and surface elasticity. The unique features of the model are that it incorporates, through the second moment, the heterogeneous nature of elasticity across the nanowire cross section, and that it accounts for transverse surface-stress-induced relaxation strains. The model is validated by comparison to benchmark atomistic calculations, existing one-dimensional surface elasticity theories based on the Young–Laplace equation, and also three-dimensional surface elasticity theories that assume homogeneous elastic properties across the nanowire cross section via three examples: surface-stress-induced axial relaxation, resonant properties of unstrained, strained and top-down nanowires, and buckling of nanowires. It is clearly demonstrated that the one-dimensional Young–Laplace models lead to errors of varying degrees for all of the boundary value problems considered because they do not account for transverse surface stress effects, and it is also shown that the Young–Laplace model results from a specific approximation of the proposed formulation. The three-dimensional surface elasticity model of Dingreville et al. (2005) is found to be more accurate than the Young–Laplace model, though both lose accuracy for ultrasmall (<5nm diameter) nanowires where the heterogeneous nature of the cross section elasticity becomes important. Overall, the present work demonstrates that continuum mechanics can be utilized to study the elastic and mechanical behavior and properties of ultrasmall nanowires if surface elastic contributions to the heterogeneous flexural rigidity are accounted for.

Study of terahertz wave propagation properties in nanoplates with surface and small-scale effects

In macroscopic and even microscopic structural elements, surface effects can be neglected and classical theories are sufficient. As the structural size decreases towards the nanoscale regime, the surface-to-bulk energy ratio increases and surface effects must be taken into account. In the present work, the terahertz wave dispersion characteristics of a nanoplate are studied with consideration of the surface effects as well as the nonlocal small-scale effects. Nonlocal elasticity theory of plate is used to derive the general differential equation based on equilibrium approach to include those scale effects. Scale and surface property dependent wave characteristic equations are obtained via spectral analysis. For the present study the material properties of an anodic alumina with crystallographic of 〈111〉 direction are considered. The present analysis shows that the effect of surface properties on the flexural waves of nanoplates is more significant. It can be found that the flexural wavenumbers with surface effects are high as compared to that without surface effects. The scale effects show that the wavenumbers of the flexural wave become highly non-linear and tend to infinite at certain frequency. After that frequency the wave will not propagate and the corresponding wave velocities tend to zero at that frequency (escape frequency). The effects of surface stresses on the wave propagation properties of nanoplate are also captured in the present work.

Eigenfrequencies and quality factors of nanofilm resonators with dissipative surface stress effects

A dissipative surface stress model is adopted to study the size-dependent surface dissipation effect on the quality factor and natural frequencies of vibrating elastic nanofilms (NFs). Kirchhoff plate theory along with the classic Zener model for interior friction in the presence of an initial surface tension is employed to obtain the frequency-dependent quality factor and the differential equation of flexural motion. Resonance frequencies and quality as a function of thickness for square NFs of selected material relaxation times with simply-supported edge conditions are presented and discussed.

Surface initiated rolling contact fatigue based on the asperity point load mechanism—A parameter study

The influence of asperity size, friction and residual surface stress on surface initiated rolling contact fatigue damage was investigated using the asperity point load mechanism. A parametric study was performed in two steps, first with a classic one-parameter-at-a-time approach, then as a 2-level full factorial design. The effect on fatigue initiation, damage size and spalling life for both early and developed spalling damage was examined for a gear application. Simplified response surfaces were derived as an engineering design tool for improved spalling resistance. The parametric investigation suggested that among the investigated parameters, reduced asperity height and local asperity friction will have the largest effect on the crack initiation risk. The simulations agreed with the engineering experiences that reduced surface roughness improves rolling contact fatigue resistance and that improved lubrication lengthens spalling lives and decreases fatigue risk. With compressive residual surface stresses the predictions suggested reduced fatigue risk and substantially decreased depth of individual spalls. Finally, predictions of experimentally observed effects of changing asperity size, friction and residual surface stress on spalling further motivates the asperity point load mechanism as the source behind surface initiated rolling contact fatigue.

Effects of residual surface stress and surface elasticity on the nonlinear free vibration of nanoscale plates

This paper studies the influence of surface effects (including the residual surface stress and surface elasticity) on the nonlinear free vibrations of nanoscale plates. The motion equations are derived by using the Hamilton’s principle and solved numerically. It is found that the influence of surface effects on the normalized period of nanoscale plates becomes increasingly significant when the thickness of the plate decreases. More importantly, the influence of the surface effects on the normalized vibration period reduces if the initial amplitude of the vibration increases. This tendency is more pronounced for the Mindlin plate theory, which includes the transverse shear effect of the plates. In addition, it is found that both the positive residual surface stress and surface elasticity increase the magnitude of the vibration velocity.

A combined surface stress and magneto-optical Kerr effect measurement setup for temperatures down to 30 K and in fields of up to 0.7 T

An optical 2-beam surface stress measurement and magneto-optical Kerr-effect has been combined with a liquid helium cooled cryostat. Sample temperatures down to 30 K and magnetic fields up to 0.7 T are achieved under UHV conditions. Low temperatures are exploited to obtain the first experimental data on the surface stress change induced by the adsorption of the noble gas Xe on Pt(111). High magnetic fields and low temperatures are used to characterize the magnetic properties of Co monolayers in longitudinal and polar Kerr geometries. The effective magnetic anisotropy is extracted from hard axis magnetization loops.

Melting Temperature of Al&lt;sub&gt;2&lt;/sub&gt;O&lt;sub&gt;3&lt;/sub&gt; under Pressure

The melting temperature-pressure phase diagram [Tm(P)-P] for corundum (Al2O3) is predicted through the Clapeyron equation where the pressure-dependent volume difference is modeled by introducing the effect of surface stress induced pressure. Al2O3has been employed to test the reliability of the model, because of its important role. Al2O3has been extensively investigated because of its widely ranging industrial applications. This includes applications as a refractory material both of high hardness and stability up to high temperatures, as a support matrix in catalysis.

Size dependent surface dissipation in thick nanowires

The Timoshenko [S. S. Rao, Vibration of Continuous Systems (Wiley, New York, 2007)] beam model is used to derive the differential equations governing the free vibrations of thick nanowires (NWs) with dissipative surface stress effects. The natural frequencies are calculated as functions of NW length as well as thickness-to-length ratio, with the effects of dissipation, transverse shear deformation, and rotary inertia being included. The effects of latter two parameters are significant especially for higher modes of vibration and shorter NWs and are different from what is naively expected based on elementary mechanics for some specific dimensions. The results are also compared with the previous study using Euler-Bernoulli beam theory.

Stress-Induced Variations in the Stiffness of Micro- and Nanocantilever Beams

The effect of surface stress on the stiffness of cantilever beams remains an outstanding problem in the physical sciences. While numerous experimental studies report significant stiffness change due to surface stress, theoretical predictions are unable to rigorously and quantitatively reconcile these observations. In this Letter, we present the first controlled measurements of stress-induced change in cantilever stiffness with commensurate theoretical quantification. Simultaneous measurements are also performed on equivalent clamped-clamped beams. All experimental results are quantitatively and accurately predicted using elasticity theory. We also present conclusive experimental evidence for invalidity of the long-standing and unphysical axial force model, which has been widely applied to interpret measurements using cantilever beams. Our findings will be of value in the development of micro- and nanoscale resonant mechanical sensors.

Effect of residual surface stress on the fracture of nanoscale materials

The problem of a mode I crack in nanomaterials under a remote mechanical load is investigated. The effect of the residual surface stress on the crack surface is considered and the solutions to the crack opening displacement (COD) and the stress intensity factor (KI) are obtained. The results show that the surface effect on the crack deformation and crack tip field are prominent at nanoscale. Moreover, COD and KI are influenced by the residual surface stress not only on the surface near the crack tip region but also on the entire crack surface.

Surface stress effects may induce softening: Euler–Bernoulli and Timoshenko buckling solutions

Surface elasticity effects may be significant for small scale structures. In this paper, the effect of surface elasticity effects is investigated for the buckling of Euler–Bernoulli and Timoshenko beams. Engesser and Haringx theory of Timoshenko shear beams are studied. The surface elasticity effects are considered for small scale beam structures based on the Laplace–Young equation, which results in an equivalent distributed loading term in the beam equation. We show that these effects are explained by their non-conservative nature that can be essentially modelled as a follower tensile loading for inextensible beams. As a consequence the usual paradigm that smaller is stiffer is not necessarily found for these structural problems. The buckling small scale shear beams in presence of surface elasticity effects is studied for various boundary conditions. For clamped-free boundary conditions, we show that the buckling load is reduced compared to the one without this surface effect. This result is consistent with some recent numerical results based on surface Cauchy–Born model and with experimental results available in the literature. For other boundary conditions such as hinge–hinge and clamped–clamped boundary conditions, the results are identical to the ones already published. We explain in this paper the surprising results observed in the literature that surface elasticity effects may soften a nanostructure for some specific boundary conditions (due to the non-conservative nature of its loading application). Furthermore, self-instability is theoretically noticed for small shear beams.