We present a continuum formulation to obtain simple expressions demonstrating the effects of surface residual stress and surface elastic constants on extensional and torsional stiffnesses of isotropic circular nanorods. Unlike the case of rectangular nanorods, we show that the stiffnesses of circular nanorods also depend on surface residual stress components. This is attributed to non-zero surface curvature inherent in circular nanorods. We further analyze their asymptotic limits in the limit of the nanorod's radius approaching both zero and infinity which correspond to surface dominated and bulk dominated regimes respectively. Finally, we use the recently proposed Helical Cauchy-Born rule and perform molecular statics calculations to obtain axial force, twisting moment and stiffnesses of the tungsten nanorod. The results from molecular statics calculations match most accurately with our formulae when compared to existing formulae by others. The article will appear soon in Mathematics and Mechanics of Solids and can also be accessed at the following link: https://www.researchgate.net/publication/322015816_Effect_of_surface_ela...

You may be interested in reading our following article: http://link.springer.com/article/10.1007/s10659-016-9586-1. We show the importance of incorporating material nonlinearity for accurate simulation of nanorods and nanotubes. The linear material laws are shown to give completely erroneous results. The nonlinear material laws for nanorods were obtained using the recently proposed "Helical Cauchy-Born rule". We also discuss how surface stress affects buckling in such nanostructures.

Recently published in JMPS:

<http://dx.doi.org/10.1016/j.jmps.2012.07.009>

In the past 5 or so years, analytical studies of surface effects on the mechanical properties of nanostructures such as nanowires have been performed predominantly using one-dimensional models like the Young-Laplace model. While many such analytical studies have been performed, what has been lacking until now is a systematic study of such analytical models as compared to benchmark atomistic studies for a range of nanomechanical boundary value problems.

This work performs such a study, and also develops a new theoretical model to calculate the flexural rigidity of nanowires from three-dimensional elasticity theory that incorporates the effects of surface stress and surface elasticity. It is very similar to a seminal work by Dingreville et al <http://dx.doi.org/10.1016/j.jmps.2005.02.012>, but is different in that it incorporates, through the second moment, the heterogeneous nature of elasticity across the nanowire cross section due to the effects of free surfaces. We use this approach to study the boundary value problems of surface-stress-induced axial relaxation, transverse vibrations and buckling. The benchmark comparisons demonstrate the need for a three-dimensional continuum formulation while pointing out the errors introduced by taking a one-dimensional model.

The helix angle, chirality, and radius of helical ribbons are predicted with a comprehensive, three-dimensional analysis that incorporates elasticity, differential geometry, and variational principles. In many biological and engineered systems, ribbon helicity is commonplace and may be driven by surface stress, residual strain, and geometric or elastic mismatch between layers of a laminated composite. Unless coincident with the principle geometric axes of the ribbon, these anisotropies will lead to spontaneous, three-dimensional helical deformations. Analytical, closed-form ribbon shape predictions are validated with table-top experiments. More generally, our approach can be applied to develop materials and systems with tunable helical geometries.

This research work, Tunable Helical Ribbons, is published by Appl. Phys. Lett. 98, 011906 (2011); doi:10.1063/1.3530441.

A companion paper, Continuum Elasticity Theory Approach for Spontaneous Bending and Helicity of Ribbons Induced by Mechanical Anisotropy, has been submitted to J. Mech. Phys. Solids.

Please allow me to note that I have recently published in Philosophical Magazine a paper that presents a general approach to Gibbsian surface thermodynamics that includes a treatment of solid surfaces. It can be accessed through the following link:

http://www.informaworld.com/smpp/content~db=all~content=a792987191

If you send me your e-mail address I can send you a pdf of the "author's copy" (I cannot post it owing to copyright issues). If interested, I can also send a pdf of an "author's copy" of a more comprehensive article that is to appear in Solid State Physics.

Thanks, R.C. Cammarata

There has recently been a great deal of discussion on imechanica regarding the effects of surface stress on the resonant properties of nanostructures such as nanowires. The controversy has revolved around the strain-independent part of the surface stress, which can be shown, i.e. by Gurtin et al. APL 1976, 529-530, or by Lu et al, PRB 2005, 085405, to have no effect on the resonant frequency of the nanobeam. The reason is because in taking the moment, and differentiating the moment to get the beam equation of motion, the strain-independent part of the surface stress drops out as it is constant, while the strain-dependent (surface elastic) part survives the differentiation.

However, the corresponding analysis has not been done, to-date, for finite deformation kinematics, which is critical in nanowires due to the large surface-stress-driven deformation that the nanowires undergo, particularly for sub-20 nm cross sections. The attached paper, recently accepted in JMPS, addresses this issue for the first time. As the paper is long, Section 5 of the paper is the relevant one, where we discuss the methodology we employ and the results; additional commentary is given in the conclusion.

To summarize the key findings: (1) The residual surface stress does impact the resonant properties of nanowires under finite deformation kinematics; in fact, the effect can be comparable to or larger than the effect of the strain-dependent part of the surface stress, depending on boundary condition. (2) Knowledge of the state of deformation in the nanowires is not sufficient to predict their resonant frequencies.

I should also note that ZP Huang and co-workers have recently shown analytically that the residual surface stress does impact the elastic properties of nanostructures if finite deformation kinematics are considered; portions of this discussion are also on iMechanica.

I certainly welcome any feedback and discussion from the imechanica community regarding the results, or the approach taken to obtain the results.

Abstract of paper recently accepted for publication in Journal of Applied Physics:

The purpose of the present work is to quantify the coupled effects of surface stresses and boundary conditions on the resonant properties of silicon nanowires. We accomplish this by using the surface Cauchy-Born model, which is a nonlinear, finite deformation continuum mechanics model that enables the determination of the nanowire resonant frequencies including surface stress effects through solution of a standard finite element eigenvalue problem. By calculating the resonant frequencies of both fixed/fixed and fixed/free <100> silicon nanowires with unreconstructed {100} surfaces using two formulations, one that accounts for surface stresses and one that does not, it is quantified how surface stresses cause variations in nanowire resonant frequencies from those expected from continuum beam theory. We find that surface stresses significantly reduce the resonant frequencies of fixed/fixed nanowires as compared to continuum beam theory predictions, while small increases in resonant frequency with respect to continuum beam theory are found for fixed/free nanowires. It is also found that the nanowire aspect ratio, and not the surface area to volume ratio, is the key parameter that correlates deviations in nanowire resonant frequencies due to surface stresses from continuum beam theory.

With a shallow chemical etching the roughness with spatial frequency below a critical value grows while the roughness of higher frequency decays.

node/1312

We present a surface Cauchy-Born approach to modeling FCC metals with nanometer scale dimensions for which surface stresses contribute significantly to the overall mechanical response. The model is based on an extension of the traditional Cauchy-Born theory in which a surface energy term that is obtained from the underlying crystal structure and governing interatomic potential is used to augment the bulk energy. By doing so, solutions to three-dimensional nanomechanical boundary value problems can be found within the framework of traditional nonlinear finite element methods. The major purpose of this work is to utilize the surface Cauchy-Born model to determine surface stress effects on the minimum energy configurations of single crystal gold nanowires using embedded atom potentials on wire sizes ranging in length from 6 to 280 nm with square cross sectional lengths ranging from 6 to 35 nm. The numerical examples clearly demonstrate that other factors beside surface area to volume ratio and total surface energy minimization, such as geometry and the percentage of transverse surface area, are critical in determining the minimum energy configurations of nanowires under the influence of surface stresses.

A recent discussion here about the effect of surface stress on vibrations of microcantilever has gained some interest from our members. A few years ago, Zhigang and I looked at surface effect on buckling of a thin elastic film on a viscous layer (Huang and Suo, Thin Solid Films 429, 273-281, 2003). Although the physical phenomena (buckling vs vibrations) are different, the conclusion is quite consistent with Wei Hong and Pradeep's comments toward the end of the discussion. That is, surface stress only contributes as a residual stress and thus does not affect the buckling wavelength (frequency in space in analogy to frequency in time for vibrations).

More recently, a group at NIST has found that, for ultrathin polystyrene films (h < 40 nm) on PDMS, the buckle wavelength depends on the film thickness nonlinearly, different from a conventional theory commonly used for sandwich plates with stiff skin layers. Using the conventional theory, they found that the elastic modulus of polystyrene decreases by an order of magnitude when the film thickness changes from 100 nm to 5 nm. I worked with them and proposed an explanation based on surface effects (see Stafford et al., Macromolecules 39, 5095-5099, 2006). Here again, we found that surface stress alone does not help with the change of buckle wavelength. As pointed out in one of Zhigang's comments, we had to introduce a superficial elastic modulus. To include both surface stress and surface modulus, we assumed a surface layer of finite thickness, where the surface stress can be accounted for as a residual stress in this surface layer. By using a relatively soft surface layer of about 2 nm, we were able to reasonably reproduce the measured thickness-dependence of buckle wavelength.

It is intuitively plausible that, for ultrathin films (or cantilever beams), surface properties (not just surface stress) can affect the mechanical behavior. For polymer thin films, a 2 nm surface layer is possible based on molecular dynamics simulations of polymer density near a free surface. This leads to an observable change of buckling behavior for film thickness less than about 50 nm. For other materials (metals or Si), I expect some differences: the surface layer should be thinner (< 1 nm?); the surface modulus could be higher than bulk (hard surface), etc. For more details of our model and some general discussions about surface effects on thin film buckling, please see a pre-print here (Huang et al., J. Aerospace Engineering, to appear in January 2007); Update on December 28, 2006: now published as J. Aerospace Engineering 20, 38-44 (2007).