Inertia Gradients Research Articles

Stress and strain-inertia gradient elasticity in free vibration analysis of single walled carbon nanotubes with first order shear deformation shell theory

A gradient-enriched shell formulation is introduced in the present study based on the first order shear deformation shell model and the stress gradient and strain-inertia gradient elasticity theories are used for dynamic analysis of single walled carbon nanotubes. It provides extensions of the first order shear deformation shell formulation with additional higher-order spatial derivatives of strains and stresses. The higher-order terms are introduced in the formulation by using the Laplacian of the corresponding lower-order terms. The proposed shell formulation includes two length scale size parameters related to the strain gradients and inertia gradients. The effects of the transverse shear, aspect ratio, circumferential and half-axial wave numbers and length scale parameters on different vibration modes of the single-walled carbon nanotubes are elucidated. The results are also compared with those obtained from a classical shell theory with Sanders–Koiter strain-displacement relationships.

Vibration analysis of single-walled carbon nanotubes using different gradient elasticity theories

The present work aims at investigating the vibrational characteristics of single-walled carbon nanotubes (SWCNTs) based on the gradient elasticity theories. The small-size effect, which plays an essential role in the dynamical behavior of nanotubes, is captured by applying different gradient elasticity theories including stress, strain and combined strain/inertia ones. The theoretical formulations are established based upon both the Euler–Bernoulli and the Timoshenko beam theories. To validate the accuracy of the present analysis, molecular dynamics (MDs) simulations are also conducted for an armchair SWCNTs with different aspect ratios. Comparisons are made between the aforementioned different gradient theories as well as different beam assumptions in predicting the free vibration response. It is shown that implementation of the strain gradient elasticity by incorporating inertia gradients yields more reliable results especially for shorter length SWCNTs on account of two small scale factors corresponding to the inertia and strain gradients. Also, the difference between two beam models is more prominent for low aspect ratios and the Timoshenko beam model demonstrates a closer agreement with MD results.

Various gradient elasticity theories in predicting vibrational response of single-walled carbon nanotubes with arbitrary boundary conditions

This article deals with the development and use of different gradient beam theories in order to predict the vibrational behavior of single-walled carbon nanotubes (SWCNTs). To address the problem of free vibration, the Euler–Bernoulli and Timoshenko beam theories in conjunction with the gradient elasticity theories including stress, strain and combined strain/inertia are implemented. The generalized differential quadrature method is employed to numerically solve the problem which can treat various boundary conditions. The results generated from the present gradient models are compared with those from molecular dynamics simulations as a benchmark of good accuracy and the proper values of small length scales used in the gradient models are proposed. This study shows prominent differences between various gradient models when the nanotube becomes very short (for aspect ratios of approximately lower than six). It is indicated that applying the strain gradient elasticity by incorporation of inertia gradients yields more reliable results especially for shorter length SWCNTs on account of two small-scale factors related to the inertia and strain gradients. Moreover, since with the reduction in the aspect ratio of nanotubes the effects of boundary conditions become dominant, a discussion is given to investigate the influence of end conditions on the vibrational characteristics of SWCNTs.

Vibration analysis of nanotubes conveying fluid based on gradient elasticity theory

In this paper two theoretical beam models are proposed for the vibration analysis of fluid-conveying nanotubes using the theory of strain gradient elasticity combined with inertia gradients. For comparison purposes, two stress gradient elasticity beam models were also discussed. Both theories were formulated using either the Euler-Bernoulli or the Timoshenko assumptions. Unlike the stress gradient beam models, in which only one material length scale parameter is introduced, the combined strain/inertia gradient beam models include two material length scale parameters related to the inertia and strain gradients, which enable us to investigate the size effect on the dynamical behavior of nanotubes conveying fluid. Results show that the size effect predicted by stress gradient beam models is not pronounced. However, for combined strain/inertia gradient beam models, the natural frequencies obtained display size-dependent properties. For small aspect ratios, the natural frequencies predicted by the combined strain/inertia gradient beam models are much smaller than the stress gradient results. Moreover, the critical flow velocities predicted by the combined strain/inertia gradient beam models are slightly higher than those predicted by the stress gradient beam models, showing that the stability of nanotubes is enhanced due to the consideration of inertia gradients.

Terahertz wave propagation in uniform nanorods: A nonlocal continuum mechanics formulation including the effect of lateral inertia

The dynamic testing of materials and components often involves predicting the propagation of stress waves in slender rods. The present work deals with the analysis of the wave propagation characteristics of nanorods. The nonlocal elasticity theory and also the lateral inertia are incorporated into classical/local rod model to capture unique features of the nanorods under the umbrella of continuum mechanics theory. The strong effect of the nonlocal scale has been obtained which leads to substantially different wave behaviors of nanorods from those of macroscopic rods. Nonlocal rod/bar model is developed for nanorods including the lateral inertia effects. The analysis shows that the wave characteristics are highly over estimated by the classical rod model, which ignores the effect of small-length scale. The wave propagation properties of the nanorod obtained from the present formulations are compared with the continuum rod model, nonlocal second and fourth order strain gradient models, Born-K a ´ rm a ´ n model and the nonlocal stress gradient model. It has also been shown that, the unstable second order strain gradient model can be replaced by considering the inertia gradient terms in the formulations. The effects of both the nonlocal scale and the diameter of the nanorod on spectrum curves are highlighted in the present manuscript. The results provided in this article are useful guidance for the study and design of the next generation of nanodevices that make use of the wave propagation properties of single-walled carbon nanotubes.

Gradient elasticity and flexural wave dispersion in carbon nanotubes

Higher-order elasticity theories have recently been used to predict the dispersion characteristics of flexural waves in carbon nanotubes (CNTs). In particular, nonlocal elasticity and gradient elasticity (with unstable strain gradients) have been employed within the framework of classical Euler-Bernoulli or improved Timoshenko beam theory to capture the dynamical behavior of CNTs. Qualitative agreement with the predictions of related molecular-dynamics (MD) simulations was observed, whereas the MD results departed significantly from those obtained with classical elasticity calculations. The present contribution aims to alert that the aforementioned higher-order models may yield questionable results for the higher wave numbers. As an alternative, gradient elasticity (with stable strain gradients), by also incorporating inertia gradients for dynamical applications, is used in combination with both Euler-Bernoulli and Timoshenko beam theories and shown to describe flexural wave dispersion in CNTs realistically for the small-to-medium range of wave numbers, i.e., the range for which MD results are available.

Urban Heat Island Modeling in Conjunction with Satellite-Derived Surface/Soil Parameters

Although it has been studied for over 160 years, the urban heat island (UHI) effect is still not completely understood, yet it is increasingly important. The main purpose of this work is to improve UHI modeling by using AVHRR (Advanced Very High Resolution Radiometer) satellite data to retrieve the surface parameters (albedo, as well as soil thermal and moisture properties). In this study, a hydrostatic three-dimensional mesoscale model was used to perform the numerical modeling. The Carlson technique was applied to retrieve the thermal inertia and moisture availability using the thermal AVHRR channels 4 and 5. The net urban effect was determined as the difference between urban and nonurban simulations, in which urban parameters were replaced by rural parameters. Two winter days were each used for two numerical simulations: a control and an urban-to-rural replacement run. Moisture availability values on the less windy day showed generally a south to north gradient downwind of the city and urban values less than rural values (the urban dry island day). Moisture availability was higher on the windy day, with uniform values in the rural and urban areas (uniform soil moisture day). The only exceptions were variations in the rural hills north of the city and the low rural values under the polluted urban plume downwind of the city. While thermal inertia values showed no urban‐rural differences on the uniform soil moisture day, they exhibited larger values over Atlanta than in surrounding rural area on the (less moist) dry island day. Two puzzling facts exist in the data: 1) lack of a north‐south thermal inertia gradient on the dry soil day to correspond to its abovementioned moisture availability gradient and 2) rural thermal inertia values do not change between both days in spite of their large difference in soil moisture. The observed lack of corresponding urban change is expected, as its thermal inertia values depend more on urban building materials than on moisture of soil. In both cases both the 2-m and surface skin UHIs showed positive values at night and negative values (an urban cool island, UCI) during the day. The larger nighttime 2-m UHI was on the dry day (0.88 vs 0.68C), while the larger daytime 2-m UCI was on the moist soil day (20.38 vs 20.58C). Note that the surface differences were almost always greater than the 2-m differences. These day‐night differences imply a rural thermal inertia lower than its urban values on both days, which is in conflict with the observations on the wet uniform soil moisture day. On the uniform thermal inertia day (wet day), both the UHI and UCI amplitudes should be less than on the other day, but this is not the case. A possible explanation for both of these conflicts is the improper influence of the urban plume on this day on lowering the thermal inertia and moisture availability values used in the replacement urban simulation.

Device for Electrical Conductivity Measurements of Glasses during Heating and Cooling

A system has been developed for electrical conductivity measurements of molten glasses in which electrode effects and nonuniform sample temperatures have been minimized. The sample shape is a small thin disk held between flat platinum electrodes to give low thermal inertia and minimum temperature gradients. Controlled heating and cooling are provided by a programmed dc power supply. Conductivity and temperature are measured periodically and punched on a paper tape which serves as the computer input. All calculations are performed in the computer and the output is logσ vs 1/T and activation energy vs temperature. Conductivities of 10−1 to 10−6 (Ω·cm)−1 have been measured in the temperature range of 1000 to 300°C.