Interaction of two nearby CNTs/nanovoids embedded in a metal matrix using modified nonlocal elasticity

Interaction of Carbon Nanotubes (CNTs) and nanovoids embedded in a metal matrix is studied comparatively. For this purpose, two nearby CNTs/nanovoids are modeled as two similar cylindrical inclusions/holes in an infinite matrix. The nonlocal stresses around the CNTs/nanovoids are obtained by applying the integral constitutive equation of the nonlocal elasticity to the stresses from the complex stress potential method. Also, in order to bring different nonlocality effects of dissimilar media into account, the influence function of the nonlocal elasticity is modified. Effects of the CNTs/nanovoids size and distance as well as far-field loading ratio on the stress state in the matrix are investigated. The CNTs strengthening effect on the matrix is shown through lower stresses at the CNTs and matrix interface in comparison to the applied stress. Furthermore, it is indicated that the modified nonlocal analysis brings out the interaction of CNTs/nanovoids at smaller distances in comparison with those the classical analysis used to predict. In other words, higher volume fraction of the CNTs/nanovoids can be incorporated in the metal matrices without devastating effects of their interactions.     

Electro-thermo-mechanical buckling of DWBNNTs embedded in bundle of CNTs using nonlocal piezoelasticity cylindrical shell theory

Axial buckling analysis of double-walled Boron Nitride nanotubes (DWBNNTs) embedded in an elastic medium under combined electro-thermo-mechanical loadings is presented in this article. Virtual displacement method based on nonlocal cylindrical piezoelasticity continuum shell theory is employed to derive the equilibrium equations. Boron Nitride nanotube (BNNT) is assumed to be surrounded by a bundle of carbon nanotubes (CNTs) as elastic medium for reinforcement. The elastic medium is simulated as Winkler–Pasternak foundation, and adjacent layers interactions are assumed to have been coupled by van der Walls (vdW) force evaluated based on the Lennard–Jones model. The effects of parameters such as electric and thermal loads, elastic medium and small scale are investigated on the buckling behavior of the DWBNNTs. The electric field and its direction are found to have affected the magnitude of the critical buckling load. Moreover, an analysis is carried out to estimate the nonlocal critical electro-thermo-mechanical load for the axial buckling of embedded DWBNNTs.     

Interaction of a dislocation with a surface crack in nonlocal elasticity

The interaction of a screw dislocation with a surface crack is analyzed in nonlocal elasticity under applied shear loading. The stress field is determined near the crack and the dislocation. The classical singularities at the crack tip and the dislocation core are eliminated. A maximum of the shear stress near the dislocation and a minimum between the crack and the dislocation are found, which reveals the interaction of the dislocation with the crack. Two particular cases are obtained: one is no dislocation and the other is no crack in the medium. Numerical calculations show the results from nonlocal elasticity are more rational physically than those from the classical one.     

Interaction of a circular second-phase particle and a screw dislocation in nonlocal elasticity

Summary The stress field, interaction energy and image force of a screw dislocation near a circular second-phase particle in a matrix are investigated analytically in nonlocal elasticity. With the result obtained we find the nonlocal surface residual on the interface between the second particle and the matrix due to the interaction of the second phase particle and the screw dislocation. All classicla singulartities for stress, interaction energy and image force are eliminated. The maxima of both stress and image force are estimated.     

Vibration analysis of orthotropic graphene sheets embedded in Pasternak elastic medium using nonlocal elasticity theory and differential quadrature method

In this paper, the small scale effect on the vibration analysis of orthotropic single layered graphene sheets embedded in elastic medium is studied. Elastic theory of the graphene sheets is reformulated using the nonlocal differential constitutive relations of Eringen. Both Winkler-type and Pasternak-type foundation models are employed to simulate the interaction between the graphene sheet and surrounding elastic medium. Using the principle of virtual work the governing differential equations are derived. Differential quadrature method is employed to solve the governing differential equations for various boundary conditions. Nonlocal theories are employed to bring out the small scale effect of the nonlocal parameter on the natural frequencies of the orthotropic graphene sheets embedded in elastic medium. Further, effects of (i) nonlocal parameter, (ii) size of the graphene sheets, (iii) stiffness of surrounding elastic medium and (iv) boundary conditions on non-dimensional vibration frequencies are investigated.     

Thermo-mechanical vibration of a single-walled carbon nanotube embedded in an elastic medium based on nonlocal elasticity theory

A single-elastic beam model has been developed to analyze the thermal vibration of single-walled carbon nanotubes (SWCNT) based on thermal elasticity mechanics, and nonlocal elasticity theory. The nonlocal elasticity takes into account the effect of small size into the formulation. Further, the SWCNT is assumed to be embedded in an elastic medium. A Winkler-type elastic foundation is employed to model the interaction of the SWCNT and the surrounding elastic medium. Differential quadrature method is being utilized and numerical solutions for thermal-vibration response of SWCNT is obtained. Influence of nonlocal small scale effects, temperature change, Winkler constant and vibration modes of the CNT on the frequency are investigated. The present study shows that for low temperature changes, the difference between local frequency and nonlocal frequency is comparatively high. With embedded CNT, for soft elastic medium and larger scale coefficients (e₀a) the nonlocal frequencies are comparatively lower. The nonlocal model-frequencies are always found smaller than the local model-frequencies at all temperature changes considered.     

Vibration characteristics of embedded multi-layered graphene sheets with different boundary conditions via nonlocal elasticity

A nonlocal elastic plate model accounting for the small scale effects is developed to investigate the vibrational behavior of multi-layered graphene sheets under various boundary conditions. Based upon the constitutive equations of nonlocal elasticity, derived are the Reissner–Mindlin-type field equations which include the interaction of van der Waals forces between adjacent and non-adjacent layers and the reaction from the surrounding media. The set of coupled governing equations of motion for the multi-layered graphene sheets are then numerically solved by the generalized differential quadrature method. The present analysis provides the possibility of considering different combinations of layerwise boundary conditions in a multi-layered graphene sheet. Based on exact solution, explicit expressions for the nonlocal frequencies of a double-layered graphene sheet with all edges simply supported are also obtained. The results from the present numerical solution, where possible, are indicated to be in excellent agreement with the existing data from the literature.     

Analysis of flexoelectric response in nanobeams using nonlocal theory of elasticity

This paper is concerned with the derivation of exact solutions for the static responses of simply supported nonlocal flexoelectric nanobeams. Considering both the direct and the converse flexoelectric effects, and employing the nonlocal theory of elasticity, the governing equations and the associated boundary conditions of the beams are derived to obtain the exact solutions for the displacements, nonlocal stresses and the electric potential in the beams. Both the direct and the converse flexoelectric effects are influenced by the nonlocal parameter. Active beams significantly counteract the applied mechanical load by virtue of the converse flexoelectric effect. The normal and the transverse shear deformations in the beams are affected by the converse flexoelectric effect resulting in the coupling of bending and stretching deformations in the beams even if the beams are homogeneous. Because of the consideration of the nonlocal theory of elasticity, the nonlocal stiffness of the beam appears to be less than the classical stiffness of the beam. The nonlocal elasticity does not influence the stresses of the passive beam while the nonlocal stresses in the active beam due to converse flexoeletric effect are less than the local or classical stresses in the active beam. The benchmark results presented here may be useful for verifying the numerical model and experimental results for nonlocal flexoelectric nanobeams. The present study suggests that the flexoelectric nanobeams may be effectively exploited for developing advanced smart nanosensors and nanoactuators. The research work carried out here also conveys that the nonlocal theory of elasticity must be employed for accurate analysis of flexoelectric solids.     

Vibration analysis of embedded nanotubes using nonlocal continuum theory

Vibration of nanotubes embedded in an elastic matrix is investigated by using the nonlocal Timoshenko beam model. Both a stress gradient and a strain gradient approach are considered. The Hamilton’s principle is adopted to obtain the frequencies of the nanotubes. The dependencies of frequency on the stiffness and mass density of the surrounding elastic matrix, the nonlocal parameter, the transverse shear stiffness and the rotary inertia of the nanotubes are obtained. The results show a significant dependence of frequencies on the surrounding medium and the nonlocal parameter. The frequencies are over-predicted by using the Euler beam model that neglects the shear stiffness and rotary inertia of the nanotubes. It is also found that the lower bound and the upper bound for the frequencies of nanotubes are, respectively, provided by the strain gradient model provides and the stress gradient theory. Explicit formulas for the frequency are obtained and therefore are easy to use by material scientists and engineers for the design of nanotubes and nanotubes based composites.     

Buckling of circular/annular Mindlin nanoplates via nonlocal elasticity

The present study proposes an analytical solution for the axisymmetric/asymmetric buckling analysis of moderately thick circular/annular Mindlin nanoplates under uniform radial compressive in-plane load. In order to consider small-scale effects, nonlocal elasticity theory of Eringen is employed. To ensure the efficiency and stability of the present methodology, the results are compared with other ones presented in the literature. Further the exact closed-form solution is obtained using three potential functions. In addition, the effect of small scales on buckling loads for different parameters such as geometry of the nanoplate, boundary conditions, and axisymmetric/asymmetric mode numbers, is investigated. It is observed that the buckling mode shape for annular nanoplates, which corresponds to the lowest critical buckling load, may be axisymmetric or asymmetric depending on boundary conditions, inner to outer radius ratios, and thickness of the nanoplate. In other words, for stiffer boundary conditions and smaller inner to outer radius ratios, the mode shape corresponding to the lowest critical buckling load is an asymmetric mode. Also, the difference between axisymmetric and asymmetric buckling loads for higher mode numbers, greater thickness to outer radius ratios and smaller outer radii decreases by increasing the nonlocal parameter.     

Torsional vibration analysis of double walled carbon nanotubes using nonlocal elasticity

In the present study, the torsional vibration behavior of double walled carbon nanotubes (DWCNTs) is investigated using nonlocal elasticity theory. The effects of van der Waals Force interaction, nanotube length and nonlocal parameter are studied in detail. Two frequency set are obtained for DWCNTs for a given half wave number. It is also shown that some mode shapes are anti-phase and some of them are in-phase. The present results can be useful in design of nano electromechanical systems like nanobearings and rotary servomotors.     

Antiplane eigenstrain problem of a circular inclusion in nonlocal elasticity

Summary The antiplane eigenstrain problem of a circular inclusion is investigated in nonlocal elasticity. The nonlocals tress fields are obtained analytically by generalizing the classical line force concept to the case of nonlocal elasticity. The nonlocal effects, stress concentration and nonlocal surface residual, are exhibited for the first time.     

Bending vibrations of rotating nonuniform nanocantilevers using the Eringen nonlocal elasticity theory

The natural frequencies of the flapwise bending vibrations of a nonuniform rotating nanocantilever has been calculated, considering the true spatial variation of the axial force due to the rotation. The area of the nanobeam cross-section is assumed to change linearly. The problem has been formulated using the nonlocal Eringen elasticity theory and it was solved by a pseudo-spectral collocation method based on Chebyshev polynomials. The effect of the nonlocal small-scale, angular speed, nonuniformity of the section and hub radius on the vibration behavior of the nanocantilever is discussed.     

Surface effects on free vibration of piezoelectric functionally graded nanobeams using nonlocal elasticity

Free vibration of functionally graded material (FGM) nanobeams is investigated by considering surface effects including surface elasticity, surface stress, and surface density as well as the piezoelectric field using nonlocal elasticity theory. The balance conditions between the nanobeam bulk and its surfaces are satisfied assuming a cubic variation for the normal stress, ${\sigma_{zz}}$ , through the piezoelectric FG nanobeam thickness. Accordingly, the surface density is introduced into the governing equation of the free vibration of nanobeams. The results are obtained for various gradient indices, voltage values of the piezoelectric field, nanobeam lengths, and mode numbers. It is shown that making changes to voltage values and modifying mechanical properties of piezoelectric FGM nanobeams are two main approaches to achieve desired natural frequencies.     

On positive definiteness of strain energy in nonlocal elasticity

The positive definiteness of strain energy in nonlocal elasticity is studied. It is shown that the strain energy of nonlocal elasticity with a continuous positive valued and monotone decreasing interaction kernel is not necessarily positive definite.     

Electron beam initiated oxidation and coalescence of metal particles embedded in a plasma-polymer thin film matrix

Microstructural changes in plasma-polymer thin films with embedded indium or silver nanoparticles were initiated by electron beam irradiation. Oxidation of indium particles was realized by electron beam irradiation in situ with the electron microscope. By means of selected area electron diffraction it was demonstrated, that indium particles oxidize to smaller In₂O₃ crystallites. On the other hand, thermal annealing of the films results only in a formation of an indium oxide shell. At plasma-polymer films with embedded silver particles, the electron beam irradiation with a microfocus electron source results in an increase of the size of the embedded silver particles with a simultaneous decrease of the particle number (coalescence).     

Vibration analysis of orthotropic graphene sheets using nonlocal elasticity theory and differential quadrature method

The small scale effect on the vibration analysis of orthotropic single layered graphene sheets (SLGS) is studied. Elastic theory of the graphene sheets is reformulated using the nonlocal differential constitutive relations of Eringen. The equations of motion of the nonlocal theories are derived for the graphene sheets. Differential quadrature method (DQM) is employed to solve the governing differential equations for various boundary conditions. Nonlocal theories are employed to bring out the small scale effect of the nonlocal parameter on the natural frequencies of the orthotropic graphene sheets. Further, effects of (i) nonlocal parameter, (ii) size of the graphene sheets, (iii) material properties and (iv) boundary conditions on nondimensional vibration frequencies are investigated.     

Electrical conduction mechanisms of metal nanoclusters embedded in an amorphous Al2O3 matrix

We present the synthesis and electrical characterization of amorphous nanocomposite layers made of metallic nanoclusters embedded in an alumina matrix (nc-Co:Al₂O₃). The nanostructured materials were fabricated using a pulsed laser deposition (PLD)-derived method based on a nano-cluster generator coupled with a conventional PLD system for host medium co-deposition. The films were subjected to a detailed structural study carried out using high-resolution transmission electron microscopy and atomic force microscopy. The clusters inserted in the alumina matrix are metallic, well crystallized and possess an fcc structure with an average diameter centered at ∼ 2 nm. Dielectric constant and electrical conduction mechanisms of nc-Co:Al₂O₃ layers integrated in metal-insulator-metal capacitive structures were studied for different doping levels and for a broad temperature range (303-473 K). It was concluded that the dielectric constant in the films depends on the doping levels while the major electrical conduction mechanisms are best described by the space charge limited currents formalism, in which the current density J on an applied voltage V follow a power-law dependence (J ∼ Vⁿ) at applied voltages higher than ∼ 2 V. Such composite may find immediate applications as dielectric layers with controlled discharging conduction paths in Radio Frequency-Micro-Electro-Mechanical Systems capacitive structures.