1 INTRODUCTION

At nano scales, surface area to volume ratio increases to the extent that the surface stress effects can no longer be ignored as noted by (^{Miller and Shenoy, 2000}) and (^{Sun and Zhang, 2003}). This phenomenon has been observed in a number of studies and, in particular, the effect of surface tension on the properties of nano-sized structures has been noted in (^{Cuenot et al., 2004};^{Jing et al., 2006};^{Park and Klein, 2008};^{Stan et al., 2008};^{Eremeyev et al., 2009};^{Wang et al., 2010}). The reason for the surface effects is the difference in the behavior of atoms depending on whether they are close to a free surface or within the bulk of the material. Atoms close to or at the surface lead to higher stiffness and mechanical strength (^{Murdoch, 2005}). A review of the effect of surface stress on nanostructures is given by (^{Wang et al, 2011}).

Presently a number of structures at the nano-scale is being studied and these include one-dimensional nanowires and nanobeams as well as the two-dimensional nanoplates and graphene. Recent work on the effect of surface energy on the mechanical behavior of nanowires include (^{Jiang and Yan, 2010};^{Hasheminejad and Gheshlaghi, 2010};^{Lee and Chang, 2011};^{Samaei et al., 2012}). Vibrations of nanobeams with surface effects have been studied by (^{Gheshlaghi and Hasheminejad, 2011}), (^{Sharabiani and Yazdi, 2013}), (^{Hosseini-Hashemi and Nazemnezhad, 2013}), (^{Malekzadeh and Shojaee, 2013}).

High area to volume ratio of nanoplates makes them particularly susceptible to surface effects and the accuracy of solutions improves by including these effects in the governing equations. Theory of plates with surface effects has been developed in (^{Lu et al., 2006}). The effect of surface stress on the stiffness of cantilever plates was studied by (^{Lachut and Sader, 2007}). (^{Wang and Wang, 2011a}) and (^{Farajpour et al., 2013}) studied the buckling of nonlocal nanoplates and included the effects of surface energy to assess its effect on the buckling load. Several studies on the vibrations and dynamics of nanoplates were conducted taking the surface effects into account in (^{Ansari and Sahmani, 2011};^{Wang and Wang, 2011b};^{Assadi, 2013};^{Narendar and Gopalakrishnan, 2012}).

In the above studies only the average values of the material properties were used and the possibility of variations and/or inaccuracies in the data was not considered. Main drawback of the studies using deterministic material properties of the nano-sized structures is that the elastic constants and other material properties such as surface tension and the small-scale parameter often cannot be determined with a high degree of accuracy. The uncertainties in determining the values of these constants are due to a number of reasons which include processing difficulties, measurement inaccuracies, and defects and imperfections in the molecular structures, to name a few. For example experimental difficulties for making accurate measurements at the nano scale can lead to significant scatter in material data as noted by (^{Kis and Zetti, 2008}) and (^{Lee et al., 2008}).

Results obtained by neglecting the possibility of uncertainty in material properties are, in general, not reliable in the sense that the load carrying capacity of the structure may be overestimated (^{Au et al., 2003}). In the case of buckling, premature buckling may occur when these uncertainties affect the structure in a negative way. However the uncertainty in the data can be incorporated into a non-deterministic model to improve the reliability of results and to compute a conservative buckling load. In the present study this is done by convex modeling which requires that the uncertain quantities are bounded by an ellipsoid (^{Luo et al., 2009}). Examples of convex modeling applied to various engineering problems with uncertain data can be found in (^{Sadek et al., 1993};^{Adali et al., 1995};^{Qiu et al., 2009};^{Kang et al., 2011};^{Luo et al., 2011};^{Radebe and Adali, 2013}), where beams, plates and columns have been studied with respect to static and dynamic response, vibration and buckling.

Nanoplates are used in several nanotechnology applications and often subjected to in-plane loads which can lead to failure by buckling, especially considering their extremely small thickness (^{Asemi et al., 2014a},^{2014b}). Buckling behavior and sensitivity of nonlocal orthotropic nanoplates with material uncertainty have been investigated by (^{Radebe and Adali, 2014}) neglecting the surface effect. Present study extends these results to take into account the effect of surface stress in the buckling of isotropic nanoplates. The material parameters taken as uncertain are residual surface stress, surface elastic modulus, the small scale parameter of the nonlocal theory and Young's modulus. To investigate the sensitivity of the buckling load to uncertain material properties, relative sensitivity indices are defined (^{Cacuci, 2003};^{Conceição António and Hoffbauer, 2013}). The effect of uncertainty on the buckling load is studied in the numerical examples and the sensitivity to uncertainty is studied by means of contour plots.

2 NONLOCAL NANOPLATE WITH SURFACE EFFECTS

We consider a rectangular nanoplate subject to biaxial buckling loads *N _{x} * and

*N*acting in the

_{y}*x*and

*y*directions, respectively. The dimensions of the plate are specified as

*a*in the

*x*-direction and

*b*in the

*y*-direction with the plate thickness given by

*h*. The differential equation governing the buckling of the nanoplate based on nonlocal elastic theory and including the effect of surface energy is given in (

^{Wang and Wang, 2011a}) as

where *w*(*x, y*) is the deflection of the plate,,,is the small-scale parameter of the nonlocal theory and *v* is the Poisson's ratio. Uncertain quantities are denoted by the tilde symbol. The uncertain material constants are(Young's modulus),(surface elastic modulus) and(residual surface stress). For a simply supported plate, the solution is given by

The buckling load can be obtained by substituting Eq. (^{2}) into Eq. (^{1}). This computation gives (^{Wang and Wang, 2011a})

where ξ = mπ/a, ζ = nπ/b, η2 = ξ2 + ζ2, R = Ny/Nx and m, n = 1, 2, 3... By defining the constants g1 and g2 given by

the buckling load given by Eq. (^{3}) can be expressed as

Introducing the uncertainty parameters Ei, the uncertain material constants can be expressed as

where the subscript "0" denotes the nominal quantities and |εi|<<1 is the amount of uncertainty for the corresponding material property and can be positive or negative. The parameters εi are unknown and have to be determined to obtain the so-called "worst-case" buckling load which corresponds to the lowest buckling load for a given level of uncertainty. Substituting Eq. (^{6}) into Eq. (^{5}), neglecting the terms withand higher order, and keeping only the terms linear in ε*i*, we obtain

where

The expression (7) for *N _{cr} * (

*E*) can be approximated as

_{i}

by linearizing it with respect to the uncertainty parameters ε*i*. This is achieved by using the relation (1 ± ε)*c* ≅ (1*c*ε) + *O*(ε^{2)} where the superscript *c* can take positive or negative values and |ε*i*|<<1. The values of *a _{imn} * appearing in Eq. (

^{9}) are given in the Appendix.

3 CONVEX MODELING

In the present section a convex model is implemented to investigate the effect of uncertainties on the buckling load. The objective is to determine the uncertainty parameters ε*i* such that the most conservative buckling load is obtained in the presence of material uncertainties. Implementing the convex modeling , the parameters ε*i* are bounded such that they satisfy the inequality

where β is the radius of a 4-dimensional ellipsoid. As such β is a measure of the level of uncertainty and satisfies the inequality β < 1 since |ε*i*|<<1. It is known that the buckling load takes its extreme values on the boundary of the ellipsoid defined by Eq. (^{10}) (see^{Sadek et al., 1993};^{Adali et al., 1995}). Thus the inequality (10) can be replaced by the equality

to compute the "worst-case" solution. The expression (9) for *N _{cr} * is to be minimized subject to the constraint (11) to compute the constants ε

*i*and to obtain the lowest buckling load. For the constrained optimization problem, the following Lagrangian, denoted by

*L*, is introduced

By setting ∂*L*(*a _{imn} *, ε

*i*)/∂ε

*i*= 0 and using Eq. (

^{11}), the parameters ε

*i*and the Lagrange multiplier λ are computed as

Thus

where the plus and minus signs correspond to the lowest and highest buckling loads.

4 SENSITIVITY ANALYSIS

The sensitivity of the buckling load to uncertain parameters can be investigated by defining relative sensitivity indices *S _{K} *(ε

*i*) given by

which is normalized with respect to the deterministic buckling load *N _{cr} *(0). In Eq. (

^{15}), the sensitivities of the buckling load to,,andwith respect to uncertainty parameters ε

*= 1, 2, 3, 4 are denoted by*

_{i}, i*S*

_{ε}(ε

_{1}),(ε2), Sτ(ε3) and Sμ(ε4), respectively, so that the subscript K stands for the respective material property. Equation (

^{15}) indicates that the buckling pressure has zero sensitivity for ε1 = 0 corresponding to the deterministic case as expected. The sensitivities SK(ε1) can be computed from Eqs. (

^{9}) and (

^{15}) as

noting that Ncr(0) =a0mn.

5 NUMERICAL RESULTS

The effect of uncertainties in the material properties on the buckling load and the sensitivity of the buckling load to the level of uncertainty are numerically studied in the present section. The results are given for a square silver nanoplate of thickness h = 5nm. The nominal values of the elastic constants are taken as E0 = 76 Gpa, v = 0.3,= 1.22 N/m and τ_{0} = N/m which are the values used in (^{Wang and Wang, 2011a}). The buckling load is normalized with respect to the buckling load *N _{L} * of a plate without surface and nonlocal effects and having constants corresponding to the nominal values of the material properties. The buckling load

*N*can be obtained from Eq. (

_{L}^{7}) by setting= τ

_{0}= μ

_{0}= ε

*i*= 0. Thus the normalized buckling load is given by

*N*(ε

_{R}= N_{cr}*i*)/

*N*.

_{L}
Figure 1shows the curves of *N _{R} * plotted against the length

*a*for various uncertainty levels with μ

_{0}= 2 nm under the biaxial loading

*N*=

_{x}*N*(

_{y}*R*= 1) It is noted that the buckling load curve for the deterministic case (β = 0.0) corresponds to the result given inFigure 1of (

^{Wang and Wang, 2011a}). The corresponding results for the uniaxial loading with

*N*(

_{y }=*R*= 0) are given inFigure 2. Both figures show that the buckling load decreases as the uncertainty in the material properties increases.

The sensitivity results are given inFigure 3with respect to,, and inFigure 4with respect toandwhich show the contour plots of the sensitivity indices given by Eq. (^{16}) plotted against the level of uncertainty and the length *a* for a square nanoplate with *N _{y } */

*N*= 1. In all cases the sensitivity of the buckling load increases with increasing uncertainty.Figure 3aindicates that the buckling load is most sensitive to changes inand least sensitive to changes in the surface modulus(Figure 3b). The second most sensitivity is observed towards the uncertainty in the small-scale parameter(Figure 4b). Sensitivities toand(Figs. 3band

_{x}^{4b}) increase as the size of the nanoplate becomes smaller, i.e., as

*a*→ 0. On the other hand the sensitivity towards the residual surface stressincreases as the nanoplate becomes larger (Fig. 4a).

6 CONCLUSIONS

Material properties at the nanoscale are usually known with a certain level of tolerance. The present study is directed to determining the buckling loads of nano-scale plates with material uncertainties and including surface stress the effect of which cannot be neglected for nanoplates. The uncertain parameters are the Young's modulus, surface elastic modulus, residual surface stress and small scale parameter of the nonlocal theory. The effect of uncertainty on the buckling load is studied and the sensitivity of the buckling load to uncertain parameters is investigated. The uncertainty is taken into account by convex modeling which determines the worst-case combination of material properties to determine the lowest buckling load for a given level of uncertainty.

In the present case convex modeling leads to a four-dimensional ellipsoid bounding the uncertain parameters and the method of Lagrange multipliers is employed to compute these parameters. Moreover sensitivity indices are defined to investigate the relative sensitivity of the buckling load to uncertainties in the elastic constants. The numerical results show the effect of increasing uncertainty on the buckling load for biaxial and uniaxial buckling loads (Figures 1and^{2}). The sensitivity studies indicate that the buckling load is most sensitive to uncertainty in Young's modulus and the size of the nanoplate affects various sensitivities in different ways. It is observed that the sensitivities to surface elastic modulus and small-scale parameter decrease (Figures 3band^{4b}) and the sensitivity to residual surface stress increases (Figure 4a) as the nanoplate becomes larger. Sensitivity to Young's modulus is mostly influenced by the level of uncertainty (Figures 3a).