Accurate estimates of the orthotropic properties of nano-materials are usually not available due to the difficulties in making measurements at nano-scale. However the values of the elastic constants may be known with some level uncertainty. In the present study an ellipsoidal convex model is employed to study the biaxial buckling of a rectangular orthotropic nanoplate with the material properties displaying uncertain-but-bounded variations around their nominal values. Such uncertainties are not uncommon in nano-sized structures and the convex analysis enables to determine the lowest buckling loads for a given level of material uncertainty. The nanoplate considered in the present study is modeled as a nonlocal plate to take the small-size effects into account with the small-scale parameter also taken to be uncertain. Method of Lagrange multipliers is applied to obtain the worst-case variations of the orthotropic constants with respect to the critical buckling load. The sensitivity of the buckling load to the uncertainties in the elastic constants is also investigated. Numerical results are given to study the effect of material uncertainty on the buckling load.
In the deterministic analysis, variations in the material properties are neglected and the average values of the elastic constants are used to obtain a mean value for the structural response. This approach does not take the deviations from the average into account even though it is usually difficult to determine the properties of a material with any certainty. This is more so for nano-sized structures which exhibit large variations in their material properties due to defects and imperfections in their molecular structures. Moreover experimental difficulties in making accurate measurements at the nano-scale lead to significant scatter in the values of elastic constants. For example the values of Young’s modulus of carbon nanotubes have been reported between 1 and 5 TPa in the literature [1], [2] and [3].
Nanoplates, made of mono or multilayer graphene, are often employed in nanotechnology applications as sensors [4] and actuators [5] as well as in many other capacities and their usage is expected to increase [6]. Quite often they are subject to in-plane loads making them susceptible to buckling due to their extremely small thickness measured in nanometers. This situation has led to several studies on the subject and the buckling of single layer graphene has been studied in [7] without taking small-scale effects into account and in [8] and [9] employing the nonlocal theory. Buckling of isotropic nanoplates has been studied in [10] and [11] and orthotropic plates in [12], [13], [14], [15], [16] and [17] employing nonlocal constitutive relations and taking various effects such as nonuniform thickness [10], temperature [13], shear deformation [14] and nonuniform in-plane loads [15] into account. Variational principles for vibrating multi-layered orthotropic graphenes sheets were given in [16] and [17]. Studies on the vibrations of orthotropic nanoplates include [18], [19], [20] and [21]. The nonlocal theory developed in the 1970s [22] and [23] includes the small-scale effects by expressing stress as a function of strain at all points of the continuum.
Buckling and vibration results given in [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20] and [21] for graphene and nanoplates are based on the deterministic values of the elastic constants and as such neglect the variations in the material properties even though such variations are common. Nominal buckling load, corresponding to a deterministic model, could be higher than the applied compressive loads, indicating a safe design. However, a safe design based on deterministic material values is not robust due to inherent uncertainties in elastic constants [24]. This situation necessitates taking the data uncertainties into account in a non-deterministic model which will improve the reliability of the results by providing conservative load-carrying estimates.
Such a model could be probabilistic or statistical requiring information on the probability distributions of random variables. Obtaining this information in many cases is a difficult task. However, data on upper and lower bounds of uncertain parameters may be known or can be estimated with reasonable accuracy in which case an approach based on convex modeling would yield the reliable results. In this case the total level of uncertainties is bounded by an n-dimensional ellipsoid where the number of dimensions is equal to the number of uncertain parameters [25]. Examples of convex modeling applied to engineering problems with uncertain data include [24], [25], [26], [27], [28], [29], [30], [31] and [32]. A comparison of convex modeling with probabilistic methods is given in [33] and the book by Ben-Haim and Elishakoff [34] details the techniques of convex modeling.
The present study involves the computation of the buckling load of an orthotropic nanoplate in the presence of material uncertainties using convex modeling with the L2 norm of the uncertainties bounded. The constitutive relations are based on the nonlocal theory of plates which takes nano-scale effects into account. The sensitivity of the critical load to uncertainty is also investigated by defining relative sensitivities in terms of uncertainty parameters [35] and [36]. Further information on sensitivity indices can be found in [37], [38] and [39]. Numerical results are given to investigate the effect of uncertainty on the buckling loads and the dependence of the relative sensitivities on the aspect ratio is studied by means of contour plots.
The effect of variations in the material properties of a rectangular orthotropic nanoplate have been studied with respect to the critical buckling load. Small-scale effect was taken into account by employing the nonlocal theory for the governing equation. The uncertain quantities were identified as the elastic constants and the small-scale parameter and treated as uncertain-but-bounded quantities. Convex modeling of the uncertainties led to a five-dimensional ellipsoid bounding the uncertainties and the method of Lagrange multipliers were implemented in obtaining the least favorable solution, i.e., the most conservative buckling load given the bound on the uncertainties. Numerical results are given for various levels of uncertainty.
