Size Effects In Nanostructures Research Articles

Exact closed-form solutions for the free vibration analysis of multiple cracked FGM nanobeams

Exact closed-form solutions for the free vibration analysis of multiple cracked FGM nanobeams with arbitrary boundary conditions based on the Nonlocal Elasticity Theory (NET) and the Timoshenko beam theory are presented. The NET considering the size effect of nanostructures is applied. FGM properties vary nonlinearly along the height of the beam. A crack model using two springs with stiffness depending on the crack depth is applied. The proposed solutions are provided explicitly as functions of integration constants determined from the standard boundary conditions. Frequency equations, that established in the form of the determinant of the matrix 3 × 3 order for arbitrary boundary conditions, are applied to analyse free vibration. Expressions for the mode shapes are given explicitly. The effects of geometry, material, nonlocal and crack parameters on the free vibration of the multiple cracked nanobeam are then analysed in detail.

Nonlinear deformation analysis of magneto-electro-elastic nanobeams resting on elastic foundation by using nonlocal modified couple stress theory

This article investigates a comprehensive analysis of the bending problem pertaining to magneto-electro-elastic (MEE) nanobeams on the Winkler-Pasternak foundation. Taken into account the influence of size effects of nano structures, a non-classical mechanical model is established to describe the bending behavior for MEE nanobeams. In this research, the nonlocal modified couple stress theory is utilized to accurately capture both the softening and hardening effects resulting from the size effect. The theory of third-order shear deformation and von Karman geometric nonlinear theory are employed in conjunction with the introduction of Maxwell's equation to develop the model of MEE nanobeam, the control equation of the nonlinear bending problem of MEE nanobeams is obtained by Minimum potential energy principle, and it is solved by Galerkin method. Finally, this study provides a detailed discussion on the influences of the ratio of nonlocal parameters and length-scale parameters, Winkler-Pasternak coefficient, external magnetic potential, external voltage and span-thickness ratio on nonlinear deflection of nanobeam.

Vibration of two-phase local/nonlocal Timoshenko nanobeams with an efficient shear-locking-free finite-element model and exact solution

According to paradoxical behaviors of common differential nonlocal elasticity, employing the two-phase local/nonlocal elasticity, to consider the size effects of nanostructures, has recently attracted the attentions of nano-mechanics researchers. Now, due to more complexity of the two-phase elasticity problems than the differential nonlocal ones, it is essential to achieve efficient methods for studying the mechanical characteristics of two-phase nanostructures. Therefore, in this work, the exact solution corresponding to the vibrations of two-phase Timoshenko nanobeams is provided for the first time. Furthermore, the shear-locking problem is investigated in the case of two-phase finite-element method (FEM), and since the FE model of local/nonlocal nanobeam is more complex than the classic one, due to coupling of all elements together, one of the main aims of the present work is to create an efficient locking-free local/nonlocal FEM with a simple and efficient beam element. To extract the exact natural frequencies, the basic form of two-phase elasticity is replaced with the equal differential equation and the obtained higher-order governing equations are solved by satisfying additional constitutive boundary conditions. To construct the two-phase FE model, an efficient and simple shear-locking-free Timoshenko beam element is introduced, and next, basic form of two-phase elasticity including local and integral form of nonlocal elasticity is utilized. No need for satisfying higher-order boundary conditions, shear-locking-free, simple shape functions and well convergence are advantages of the present two-phase finite element model. Several convergence and comparison studies are conducted, and the reliability and locking-free properties of the present two-phase finite element model are confirmed. Also, the influences of two-phase elasticity on the natural frequencies of Timoshenko nanobeams with different thickness ratios are studied.

Nonlocal strain gradient torsion of elastic beams: variational formulation and constitutive boundary conditions

Nonlocal strain gradient continuum mechanics is a methodology widely employed in literature to assess size effects in nanostructures. Notwithstanding this, improper higher-order boundary conditions (HOBC) are prescribed to close the corresponding elastostatic problems. In the present study, it is proven that HOBC have to be replaced with univocally determined boundary conditions of constitutive type, established by a consistent variational formulation. The treatment, developed in the framework of torsion of elastic beams, provides an effective approach to evaluate scale phenomena in smaller and smaller devices of engineering interest. Both elastostatic torsional responses and torsional free vibrations of nano-beams are investigated by applying a simple analytical method. It is also underlined that the nonlocal strain gradient model, if equipped with the inappropriate HOBC, can lead to torsional structural responses which unacceptably do not exhibit nonlocality. The presented variational strategy is instead able to characterize significantly peculiar softening and stiffening behaviours of structures involved in modern Nano-Electro-Mechanical-Systems (NEMS).

Transferable screened range-separated hybrids for layered materials: The cases ofMoS2and h-BN

Screened range-separated hybrid (SRSH) functionals are of potential interest as a computationally inexpensive yet accurate alternative approach for studying (opto)electronic properties in the solid state. At present, SRSH functionals are typically tuned to reproduce with high accuracy the properties of either bulk or low-dimensional structures, rendering such functionals not only material specific, but also structure specific. The transferability of tuned SRSH functionals between bulk and low-dimensional phases has not been examined systematically and is desirable for straightforward and consistent modeling of size effects in nanostructures. We present a simple yet effective approach for simultaneous tuning of the fraction of short-range exact exchange and the range-separation parameter, which delivers accurate and transferable SRSH functionals for two-dimensional and bulk phases of two prototypical layered materials, molybdenum disulfide (${\mathrm{MoS}}_{2}$) and hexagonal boron nitride (h-BN). The ground-state SRSH band structures, resulting from minimal fitting of the SRSH to a single quasiparticle energy, are found to be in excellent agreement with GW calculations over the entire Brillouin zone. Excited-state properties are predicted using time-dependent density functional theory calculations, based on the SRSH (TD-SRSH) functional. Calculated absorption spectra are found to be in excellent agreement with GW and Bethe-Salpeter equation (BSE) calculations for ${\mathrm{MoS}}_{2}$, but less so for h-BN; the failure of the TD-SRSH for the latter material is examined and a BSE approach based on the SRSH ground state is shown to restore accuracy.

Variational nonlocal gradient elasticity for nano-beams

In the paper (Zaera, Serrano, & Fernández-Sáez, 2019) it is claimed that the nonlocal strain gradient theory (NSGT) leads to ill-posed structural problems. This conclusion was motivated by observing that Constitutive Boundary Conditions (CBC) conflict with non-standard Kinematic and Static Higher-Order Boundary Conditions (KHOBC) - (SHOBC). In the present study, it is shown that no ill-posedness holds if NSGT is established by an adequate variational formulation, with appropriate test fields. KHOBC and SHOBC have nothing to do with the proper mathematical formulation and thus they have not to be prescribed, while standard kinematic and static boundary conditions and CBC have to be imposed to close the relevant nonlocal gradient problem. This conclusion follows from a well-posed abstract variational scheme conceived for nonlocal gradient inflected beams. The treatment provides as special cases most of the size-dependent models adopted in Engineering Science to assess size-effects in nanostructures, such as NSGT, strain-driven and stress-driven local-nonlocal elasticity approaches. Additionally, a well-posed Nonlocal Stress Gradient (NStressG) model is presented, coupling the stress-driven nonlocal strategy (Romano and Barretta, 2017) with the stress gradient elasticity. The presented methodology is elucidated and validated by investigating the structural behavior of a variety of inflected nano-beams of nanotechnological interest, such as sensors and actuators. NStressG is able to predict both softening and hardening nonlocal responses and, unlike the special stress gradient elasticity model, leads to well-posed structural problems in Nanomechanics.

A Study into the Temperature and Size Effects in Nanostructures on Their Fracture under External Mechanical Loads

The paper presents a molecular dynamics study into the temperature and size effects in nanostructures on their mechanical characteristics and fracture. The study shows that among the cross-sectional areas studied, the least one measuring ny × nz = 5 × 5 lattice cells is boundary for perfect nanostructures, and at larger areas, these characteristics tend to those of macrostructures. dor all systems considered, a linear decrease with increasing temperature is observed in Young's modulus and in critical applied stress at which fracture occurs.

Time-fractional Schrödinger equation from path integral and its implications in quantum dots and semiconductors

A new fractional Schrodinger equation is constructed from path integral based on the notions of fractional velocity recently introduced in literature and the concept of fractional actionlike variational approach motivated from fractal arguments. The new equation is characterized by an emergent position-dependent mass and a time-dependent effective potential where both have important implications in semiconductors and molecular physics. Based on this equation, the problem of the quantum particle in a box is analyzed where the consequent outcomes are applied to semiconductors. It was observed that quantum size effects in nanostructures are significant and enhancement in the ground energy state for specific values of the fractional parameter is obtained. The fractional density of states for the case of bulk materials with no confinement is derived and it was revealed that a significant number of available states may be occupied even for lengths of the order of picometers. Additional results were obtained and discussed accordingly.

Phonon bottleneck identification in disordered nanoporous materials

Nanoporous materials are a promising platform for thermoelectrics in that they offer high thermal conductivity tunability while preserving good electrical properties, a crucial requirement for high- effciency thermal energy conversion. Understanding the impact of the pore arrangement on thermal transport is pivotal to engineering realistic materials, where pore disorder is unavoidable. Although there has been considerable progress in modeling thermal size effects in nanostructures, it has remained a challenge to screen such materials over a large phase space due to the slow simulation time required for accurate results. We use density functional theory in connection with the Boltzmann transport equation, to perform calculations of thermal conductivity in disordered porous materials. By leveraging graph theory and regressive analysis, we identify the set of pores representing the phonon bottleneck and obtain a descriptor for thermal transport, based on the sum of the pore-pore distances between such pores. This approach provides a simple tool to estimate phonon suppression in realistic porous materials for thermoelectric applications and enhance our understanding of heat transport in disordered materials.

Free vibrations of Bernoulli-Euler nano-beams by the stress-driven nonlocal integral model

Abstract Nonlocal theories of Continuum Mechanics are widely used in order to assess size effects in nano-structures. In this paper, free vibrations of nano-beams are investigated by making recourse to the novel stress-driven nonlocal integral model (SDM). Equations of motion governing the dynamics of a Bernoulli-Euler nano-beam are consistently formulated and numerically integrated by Matlab. Selected case studies involving structures of nanotechnological interest are examined. Natural frequencies, evaluated according to the SDM, are compared with those obtained by the Eringen differential law (EDM) and by the gradient elasticity theory (GradEla). SDM provides an effective methodology to describe nonlocal phenomena in NEMS.

Electro-mechanical vibration of nanoshells using consistent size-dependent piezoelectric theory

In this paper, the free vibrations of a short cylindrical nanotube made of piezoelectric material are studied based on the consistent couple stress theory and using the shear deformable cylindrical theory. This new model has only one length scale parameter and can consider the size effects of nanostructures in nanoscale. To model size effects in nanoscale, and considering the nanotube material which is piezoelectric, the consistent couple stress theory is used. First, using Hamilton\'s principle, the equations of motion and boundary condition of the piezoelectric cylindrical nanoshell are developed. Afterwards, using Navier approach and extended Kantorovich method (EKM), the governing equations of the system with simple-simple (S-S) and clamped-clamped (C-C) supports are solved. Afterwards, the effects of size parameter, geometric parameters (nanoshell length and thickness), and mechanical and electric properties (piezoelectric effect) on nanoshell vibrations are investigated. Results demonstrate that the natural frequency on nanoshell in nanoscale is extremely dependent on nanoshell size. Increase in size parameter, thickness and flexoelectric effect of the material leads to increase in frequency of vibrations. Moreover, increased nanoshell length and diameter leads to decreased vibration frequency.

Application of an enhanced version of the Eringen differential model to nanotechnology

Abstract Nonlocal formulations are currently adopted to assess size effects in nanostructures. The Eringen differential constitutive model, due to its simplicity, is widely used to investigate the size-dependent behavior of beams at nanoscale. Nevertheless this theory provides vanishing size effects in nanobeams under point loads and exhibits stiffening phenomena in nanocantilevers for increasing values of the nonlocal factor. Accordingly, the analysis of structures like nanoactuators is commonly carried out by using more complex constitutive approaches, such as gradient or integral models. A new strategy is outlined in the present paper as a possibly convenient alternative, by equipping the Eringen differential law with an additional term involving the derivative of the axial stress. Effectiveness of the new methodology is tested with reference to simple case studies.

Surface piezoelectricity: Size effects in nanostructures and the emergence of piezoelectricity in non-piezoelectric materials

In this work, using a combination of a theoretical framework and atomistic calculations, we highlight the concept of “surface piezoelectricity,” which can be used to interpret the piezoelectricity of nanostructures. Focusing on three specific material systems (ZnO, SrTiO3, and BaTiO3), we discuss the renormalization of apparent piezoelectric behavior at small scales. In a rather interesting interplay of symmetry and surface effects, we show that nanostructures of certain non-piezoelectric materials may also exhibit piezoelectric behavior. Finally, for the case of ZnO, using a comparison with first principles calculations, we also comment on the fidelity of the widely used core–shell interatomic potentials to capture non-bulk electro-mechanical response.