Nonlocal Stress Theory Research Articles

Photothermoelastic interaction in a two-dimensional semiconductor with nonlocal stress theory

The present contribution enlightens a viable simulation which describes the photothermoelastic interactions for a two-dimensional thermoelastic transversely isotropic thick semiconducting plate due to the presence of a varying heat source. The upper surface of the semiconductor is traction-free, subjected to prescribed surface temperature and prescribed carrier density flux while the lower surface of the plate rests on a rigid foundation and is thermally insulated. In this new model, the conventional Fourier law has been modified assimilating the memory-dependent derivative within a slipping interval in association with the nonlocal stress theory. On utilizing Laplace and Fourier transformation mechanism, the governing equations for displacement, carrier density and temperature have been obtained in the Laplace–Fourier transform domain. In order to have the solutions in the space-time domain, the inversion of the double transform have been carried out numerically based on the method of Fourier series expansion. From the graphical representations corresponding to the numerical results, the significance of effective parameters such as nonlocality parameter, delay-time parameter is discussed. Noteworthy differences in the results have been demonstrated for different kernel functions. Also, the superiority of a nonlinear kernel function and its effectiveness is also reported in the study.

Elasto‐thermodiffusive interaction under void due to nonlocal stress theory

AbstractThe current investigation address a novel generalized elasto‐thermodiffusion model for a thermoelastic porous half‐space incorporating the nonlocal stress theory proposed by Eringen. Modeling of the problem is performed by adopting Moore‐Gibson‐Thompson (MGT) thermoelasticity theory defined in an integral form of a common derivative on a slipping interval, well known as the memory‐dependent derivative. The bounding plane of the medium is subjected to time‐dependent thermal and chemical shocks and there is no change in the volume fraction field. Laplace transform and the Fourier transform techniques have been adopted to represent the analytical solutions in the transformed domain. The distributions of the physical fields such as the temperature, stress, chemical potential, mass concentration and the volume fraction field were found in the real space‐time domain adopting suitable numerical scheme based on the Fourier series expansion. According to the discussion of the computational results and the respective graphical representations, the prominent role of different parameters such as the effect of nonlocality, effect of void and thermodiffusion is analyzed. Moreover, the superiority of a nonlinear kernel function compared to a linear form is also reported.

Nonlinear Vibration of Function Graded Magneto-electro-elastic Microplates using Nonlocal Stress Theory

This work presents an analysis of nonlinear vibration in Function Graded Magneto-Electro-Elastic (FG-MEE) plates subjected to mechanical, electrical, and magnetic loads using the nonlocal stress theory approach. In this study two types of MEE plates, namely BaTiO3 and CoFe2O4 were considered. The basic equations are derived using classical plate theory with nonlocal stress theory and are solved using the Galerkin method and Runge-Kutta method. We investigated the effects of nonlocal parameters, materials, and geometrical characteristics on the natural frequencies and nonlinear vibration of the FG-MEE microplates.
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A Perturbation Approach for Lateral Excited Vibrations of a Beam-like Viscoelastic Microstructure Using the Nonlocal Theory

In this paper, we investigate the lateral vibration of fully clamped beam-like microstructures subjected to an external transverse harmonic excitation. Eringen’s nonlocal theory is applied, and the viscoelasticity of materials is considered. Hence, the small-scale effect and viscoelastic properties are adopted in the higher-order mathematical model. The classical stress and classical bending moments in mechanics of materials are unavailable when modeling a microstructure, and, accordingly, they are substituted for the corresponding effective nonlocal quantities proposed in the nonlocal stress theory. Owing to an axial elongation, the nonlinear partial differential equation that governs the lateral motion of beam-like viscoelastic microstructures is derived using a geometric, kinematical, and dynamic analysis. In the next step, the ordinary differential equations are obtained, and the time-dependent lateral displacement is determined via a perturbation method. The effects of external excitation amplitude on excited vibration are presented, and the relations between the nonlocal parameter, viscoelastic damping, detuning parameter, and the forced amplitude are discussed. Some dynamic phenomena in the excited vibration are revealed, and these have reference significance to the dynamic design and optimization of beam-like viscoelastic microstructures.

Thermo-Electro-Mechanical Size-Dependent Buckling Response for Functionally Graded Graphene Platelet Reinforced Piezoelectric Cylindrical Nanoshells

An accurate buckling response analysis for functionally graded graphene platelet (GPL) reinforced piezoelectric cylindrical nanoshells subject to thermo-electro-mechanical loadings is presented by a rigorous symplectic expansion approach. Three types of GPL reinforced patterns are considered, and the modified Halpin–Tsai model is employed to determine their effective material properties. By using Eringen’s nonlocal stress theory and Reissner’s shell theory, new governing equations are established in the Hamiltonian form. Exact solutions are expanded into symplectic series and three possible forms are derived. A comparison with the existing study is presented to validate the solution and very good agreement is observed. The effects of material and geometrical properties of GPLs, electric voltage and temperature rise on critical buckling stresses are investigated and discussed in detail.

Molecular dynamics simulation and size dependent cylindrical shell models for vibrations of spinning axially loaded carbon nanotubes

Today nanotubes are main parts of nano-machines who withstand against axial and vibrational loads. So, in this paper a mathematical model is presented to predict behavior of a spinning nanotube under axial load. For this purpose, classical thin shell theory and first order shear deformation shell theory are combined with nonlocal stress theory. The effect of rotation is considered by adding centrifugal and Coriolis forces in the formulations. Also, effect of axial compressive load is added in the formulation by considering buckling energy term. Equations are solved for both theories and numerical results are compared with the literature for validation. In the case of buckling of spinning nanotube there is not sufficient numerical results for comparison. So, Molecular dynamics (MD) simulation is done and its results in stationary and rotating conditions are compared with the mathematical model. These comparisons approved validity and accuracy of the mathematical models. Finally, effect of changing different parameters on the behavior of nanotube is investigated.

Torsional Vibration of Cracked Nanobeam Based on Nonlocal Stress Theory with Various Boundary Conditions: An Analytical Study

In this paper, the torsional vibration of cracked nanobeam was studied based on a nonlocal elasticity theory. The location of the crack is simulated by a torsional spring which links segments of nanobeam together. Also, different boundary conditions, including clamped–free, clamped–clamped and clamped–torsional spring, were considered. Furthermore, a detailed parametric study was conducted to investigate the influence of crack location, nonlocal parameter, length of nanobeam, spring constant and end supports on the torsional vibration.

A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation

In recent years there have been many papers that considered the effects of material length scales in the study of mechanics of solids at micro- and/or nano-scales. There are a number of approaches and, among them, one set of papers deals with Eringen's differential nonlocal model and another deals with the strain gradient theories. The modified couple stress theory, which also accounts for a material length scale, is a form of a strain gradient theory. The large body of literature that has come into existence in the last several years has created significant confusion among researchers about the length scales that these various theories contain. The present paper has the objective of establishing the fact that the length scales present in nonlocal elasticity and strain gradient theory describe two entirely different physical characteristics of materials and structures at nanoscale. By using two principle kernel functions, the paper further presents a theory with application examples which relates the classical nonlocal elasticity and strain gradient theory and it results in a higher-order nonlocal strain gradient theory. In this theory, a higher-order nonlocal strain gradient elasticity system which considers higher-order stress gradients and strain gradient nonlocality is proposed. It is based on the nonlocal effects of the strain field and first gradient strain field. This theory intends to generalize the classical nonlocal elasticity theory by introducing a higher-order strain tensor with nonlocality into the stored energy function. The theory is distinctive because the classical nonlocal stress theory does not include nonlocality of higher-order stresses while the common strain gradient theory only considers local higher-order strain gradients without nonlocal effects in a global sense. By establishing the constitutive relation within the thermodynamic framework, the governing equations of equilibrium and all boundary conditions are derived via the variational approach. Two additional kinds of parameters, the higher-order nonlocal parameters and the nonlocal gradient length coefficients are introduced to account for the size-dependent characteristics of nonlocal gradient materials at nanoscale. To illustrate its application values, the theory is applied for wave propagation in a nonlocal strain gradient system and the new dispersion relations derived are presented through examples for wave propagating in Euler–Bernoulli and Timoshenko nanobeams. The numerical results based on the new nonlocal strain gradient theory reveal some new findings with respect to lattice dynamics and wave propagation experiment that could not be matched by both the classical nonlocal stress model and the contemporary strain gradient theory. Thus, this higher-order nonlocal strain gradient model provides an explanation to some observations in the classical and nonlocal stress theories as well as the strain gradient theory in these aspects.

Nonlocal theoretical approaches and atomistic simulations for longitudinal free vibration of nanorods/nanotubes and verification of different nonlocal models

Recently, there have been an increasing number of studies on nonlocal theoretical models, of which different kinds of nonlocal elasticity approaches including softening and hardening models have been investigated extensively. In the present work, the longitudinal dynamic behaviors of some common one-dimensional nanostructures (e.g. nanorods/nanotubes) were examined using the hardening nonlocal approach. The hardening nonlocal longitudinal stress was employed and an eigenvalue analysis method was utilized in deriving some dynamic characteristics of the minute structures at nanoscale. The effects of a dimensionless nonlocal small scale parameter at molecular level unavailable in classical rods/tubes were investigated. Subsequently, nonlocal longitudinal vibration responses under various boundary conditions including soft and hard constraints were determined through a numerical method. The correlations between natural frequencies and the nonlocal nanoscale parameter were obtained and discussed in detail. Within the framework of hardening nonlocal stress theory, it concludes for the first time that the longitudinal free vibration frequencies of nanorods/nanotubes are higher than those based on the classical continuum mechanics but they are quite different from the softening nonlocal model. The strengthening effects on nonlocal stiffness of nanorods/nanotubes are observed and they are consistent with some other theoretical approaches or atomistic simulations. In particular, a case study on nonlocal natural frequencies of carbon nanotubes (CNTs) was presented and the results were calculated by atomistic simulations, classical continuum theory, softening and hardening nonlocal models, respectively. The validity of both existing softening and hardening nonlocal models was confirmed in comparison with the atomistic simulations. In addition, a comparative calculation for dimensional natural frequencies with respect to length of CNTs by different methodologies was provided to explain why the softening and hardening nonlocal models are both correct in nonlocal elasticity theory.

A nonlocal analytical approach for torsion of cylindrical nanostructures and the existence of higher-order stress and geometric boundaries

A new analytical approach considering the effective nonlocal shear stress field is proposed and subsequently the torsion of cylindrical nanostructures, such as nanorods, nanoshafts, and/or nanotubes is investigated. The analytical solutions and torsional characteristics of such nanostructures subjected to an external end torque or distributed torques are established based on the Eringen’s gradient type of nonlocal elasticity theory. For bending, buckling and vibration, it has been proved that the classical equations of equilibrium and/or equations of motion could not be applied to nonlocal nanostructures directly unless the stress and moment quantities in classical equations are replaced by the corresponding effective nonlocal quantities. The work is extended to derive the governing equations here via the variational principle and effective nonlocal stress theory for torsion of cylindrical nanostructures for the first time. A new strain energy density expression with higher-order nonlocal shear strain gradients and a nonlocal nanoscale parameter is established in detail, alongside the corresponding generalized non-classical stress and geometric boundary conditions. The effective nonlocal twisting moment is proposed for torsion deformation for the first time. A few benchmarked examples are solved and the analytical solutions for specific cases of interest demonstrate that the maximum angular deformation for static twist decreases with an increase in the nonlocal nanoscale parameter, or the torsion rigidity of nonlocal nanostructures tends to harden when the effective nonlocal effects are considered. The effective nonlocal torsional solutions obtained are compared with the classical and previous softening nonlocal solutions to reveal the significant effects of additional higher-order nonlocal boundary conditions. The nanostructural enhancement effect derived here is consistent with some other existing approaches including the strain gradient elasticity theory and the modified couple stress theory.

Axial buckling analysis of vertically aligned ensembles of single-walled carbon nanotubes using nonlocal discrete and continuous models

Several novel models, based on the nonlocal stress theory, on the lateral buckling of two- and three-dimensional (2D and 3D) ensembles of vertically aligned single-walled carbon nanotubes (SWCNTs) are developed. The existing van der Waals forces between the constitutive atoms of the neighboring SWCNTs are modeled by an elastic layer, and each individual SWCNT is modeled using the nonlocal Rayleigh beam theory. The governing equations for lateral buckling of both 2D and 3D SWCNTs ensembles due to axial compressive loads are derived. These are called discrete models since the lateral deformations of each SWCNT only express in terms of the longitudinal coordinate of the SWCNT. It will be revealed that the lateral deformation of SWCNTs can be also stated in terms of two and three coordinates for 2D and 3D ensembles of SWCNTs, respectively. The resulting models are called continuous models. Based on both discrete and continuous models, the critical axial buckling loads for both 2D and 3D ensembles of SWCNTs for a special case, when the outer SWCNTs are prohibited from any lateral movement, are obtained. A reasonably good agreement between the results of the discrete model and those of the continuous model is reported. Subsequently, the roles of small-scale parameter, intertube distance, and number of SWCNTs on the critical buckling loads are examined. The obtained results and the proposed models in the present work would be very helpful in design and fabrication of 2D and 3D groups of vertically aligned SWCNTs, particularly those whose main duties are to transfer securely the applied axial forces.

Cross-Scale Modeling and Optimization of a Cantilever-Type Micro/Nano Valve Based on Nonlocal Stress Theory

Cross-Scale Modeling and Optimization of a Cantilever-Type Micro/Nano Valve Based on Nonlocal Stress Theory

Free torsional vibration of nanotubes based on nonlocal stress theory

A new elastic nonlocal stress model and analytical solutions are developed for torsional dynamic behaviors of circular nanorods/nanotubes. Unlike the previous approaches which directly substitute the nonlocal stress into the equations of motion, this new model begins with the derivation of strain energy using the nonlocal stress and by considering the nonlinear history of straining. The variational principle is applied to derive an infinite-order differential nonlocal equation of motion and the corresponding higher-order boundary conditions which contain a nonlocal nanoscale parameter. Subsequently, free torsional vibration of nanorods/nanotubes and axially moving nanorods/nanotubes are investigated in detail. Unlike the previous conclusions of reduced vibration frequency, the solutions indicate that natural frequency for free torsional vibration increases with increasing nonlocal nanoscale. Furthermore, the critical speed for torsional vibration of axially moving nanorods/nanotubes is derived and it is concluded that this critical speed is significantly influenced by the nonlocal nanoscale.

Twisting statics and dynamics for circular elastic nanosolids by nonlocal elasticity theory

Abstract The torsional static and dynamic behaviors of circular nanosolids such as nanoshafts, nanorods and nanotubes are established based on a new nonlocal elastic stress field theory. Based on a new expression for strain energy with a nonlocal nanoscale parameter, new higher-order governing equations and the corresponding boundary conditions are first derived here via the variational principle because the classical equilibrium conditions and/or equations of motion can-not be directly applied to nonlocal nanostructures even if the stress and moment quantities are replaced by the corresponding nonlocal quantities. The static twist and torsional vibration of cir-cular, nonlocal nanosolids are solved and discussed in detail. A comparison of the conventional and new nonlocal models is also presented for a fully fixed nanosolid, where a lower-order governing equation and reduced stiffness are found in the conventional model while the new model reports opposite solutions. Analytical solutions and numerical examples based on the new nonlocal stress theory demonstrate that nonlocal stress enhances stiffness of nanosolids, i.e. the angular displace-ment decreases with the increasing nonlocal nanoscale while the natural frequency increases with the increasing nonlocal nanoscale.

Thermal effects on buckling of shear deformable nanocolumns with von Kármán nonlinearity based on nonlocal stress theory

Thermal buckling of nanocolumns considering nonlocal effect and shear deformation is investigated based on the nonlocal elasticity theory and the Timoshenko beam theory. By expressing the nonlocal stress as nonlinear strain gradients and based on the variational principle and von Kármán nonlinearity, new higher-order differential governing equations with corresponding higher-order nonlocal boundary conditions both in transverse and axial directions for instability of nanocolumns are derived. New analytical solutions for some practical examples on instability of nanocolumns are presented and analyzed in detail. The paper concluded that the critical buckling load is significantly increased in the presence of nonlocal stress and the results confirm that nanocolumn stiffness is enhanced by nanoscale size effect and reduced by shear deformation. The critical temperature change is increased with larger diameter to length ratio and higher nonlocal nanoscale. It is also concluded that at low and room temperatures the buckling load of nanocolumns increases with increasing temperature change, while at high temperature the buckling load decreases with increasing temperature change.

Vibration Analysis of Axially Compressed Nanobeams and its Critical Pressure Using a New Nonlocal Stress Theory

The transverse vibration of a nanobeam subject to initial axial compressive forces based on nonlocal elasticity theory is investigated. The effects of a small nanoscale parameter at molecular level unavailable in classical mechanics theory are presented and analyzed. Explicit solutions for natural frequency, vibration mode shapes are derived through two different methods: separation of variables and multiple scales. The respective numerical solutions are in close agreement. Validity of the models and approaches presented in the work are verified. Unlike the previous studies for a nonlocal nanostructure, this paper adopts the effective nonlocal bending moment instead of the pure traditional nonlocal bending moment. The analysis yields an infinite-order differential equation of motion which governs the vibrational behaviors. For practical analysis and as examples, an eight-order governing differential equation of motion is solved and the results are discussed. The paper presents a complete nonlocal nanobeam model and the results may be helpful for the application and design of various nano-electro-mechanical devices, e.g. nano-drivers, nano-oscillators, nano-sensors, etc., where a nanobeam acts as a basic element.

Nonlocal Stress Theory for Buckling Instability of Nanotubes: New Predictions on Stiffness Strengthening Effects of Nanoscales

Nonlocal Stress Theory for Buckling Instability of Nanotubes: New Predictions on Stiffness Strengthening Effects of Nanoscales

Free vibration of pre-tensioned nanobeams based on nonlocal stress theory

The transverse free vibration of nanobeams subjected to an initial axial tension based on nonlocal stress theory is presented. It considers the effects of nonlocal stress field on the natural frequencies and vibration modes. The effects of a small scale parameter at molecular level unavailable in classical macro-beams are investigated for three different types of boundary conditions: simple supports, clamped supports and elastically-constrained supports. Analytical solutions for transverse deformation and vibration modes are derived. Through numerical examples, effects of the dimensionless nanoscale parameter and pre-tension on natural frequencies are presented and discussed.

A nonlocal theory for stress in bound, Brownian suspensions of slender, rigid fibres

A nonlocal theory for stress in bound suspensions of rigid, slender fibres is developed and used to predict the rheology of dilute, rigid polymer suspensions when confined to capillaries or fine porous media. Because the theory is nonlocal, we describe transport in a fibre suspension where the velocity and concentration fields change rapidly on the fibre's characteristic length. Such rapid changes occur in a rigidly bound domain because suspended particles are sterically excluded from configurations near the boundaries. A rigorous no-flux condition resulting from the presence of solid boundaries around the suspension is included in our nonlocal stress theory and naturally gives rise to concentration gradients that scale on the length of the particle. Brownian motion of the rigid fibres is included within the nonlocal stress through a Fokker–Planck description of the fibres’ probability density function where gradients of this function are proportional to Brownian forces and torques exerted on the suspended fibres. This governing Fokker–Planck probability density equation couples the fluid flow and the nonlocal stress resulting in a nonlinear set of integral-differential equations for fluid stress, fluid velocity and fibre probability density. Using the method of averaged equations (Hinch 1977) and slender-body theory (Batchelor 1970), the system of equations is solved for a dilute suspension of rigid fibres experiencing flow and strong Brownian motion while confined to a gap of the same order in size as the fibre's intrinsic length. The full solution of this problem, as the fluid in the gap undergoes either simple shear or pressure-driven flow, is solved self-consistently yielding average fluid velocity, shear and normal stress profiles within the gap as well as the probability density function for the fibres’ position and orientation. From these results we calculate concentration profiles, effective viscosities and slip velocities and compare them to experimental data.