A staggered nonlocal multiscale model for a heterogeneous medium

A staggered nonlocal multiscale model for a heterogeneous medium is developed and validated. The model is termed as staggered nonlocal in the sense that it employs current information for the point under consideration and past information from its local neighborhood. For heterogeneous materials, the concept of phase nonlocality is introduced by which nonlocal phase eigenstrains are computed using different nonlocal phase kernels. The staggered nonlocal multiscale model has been found to be insensitive to finite element mesh size and load increment size. Furthermore, the computational overhead in dealing with nonlocal information is mitigated by superior convergence of the Newton method. Copyright © 2012 John Wiley & Sons, Ltd.     

Superfluid Mass-Energy Densities of Nonlocal Particle and Gravitational Field

Dynamical gravitational and geodesic equations are derived for superfluid densities of nonlocal self-coherent particles. The geometrized gravitational particle is the r ⁻⁴ distribution of inertial mass that balances Ricci curvatures in the Einstein equation without the right-hand side. The spatial energy integral of such an infinite radial particle is finite and determines its nonlocal gravimechanical charge for energy-to-energy interactions with other nonlocal particles. Non-empty space of the flat material world is filled continuously by overlapping energy-flows of all nonlocal particles and their fields.     

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.     

Theories of nonlocal plasticity

Deformation and rate theories of nonlocal plasticity are formulated. Constitutive equations are obtained for elasto-plastic solids extending Lévy-von Mises and Prandtl-Reuss theories to include nonlocal effects. Combined elastic-plastic constitutive equations are given. A nonlocal deformation theory is also presented. Thermodynamical restrictions are studied.     

Nonlocal continuum mechanics based on distributions

This paper presents a derivation of the balance laws of nonlocal continuum mechanics, based on the concept of microelements (very small collections of atoms or molecules). Mechanical fields (mass density, stress tensor, internal energy density, …) are assumed to be distributions (generalized functions), over a microelement surface and volume, influenced by distant microelements. Collection of these distributions in the limit when the surface and volume elements approach zero, defines the nonlocal mechanical fields. By this process, singularities and discontinuities that may be present in microelements, are smoothed out. The nonlocal balance laws are then established, having no nonlocal residuals.     

Concentrated ring loading in a nonlocal elastic medium

Assuming an appropriate nonlocal modulus and using the Boussinesq–Galerkin vector representation of the nonlocal stress field the stress distribution in a nonlocal elastic medium has been found under the concentrated ring normal and shear loadings and force dipoles. The nonlocal modulus used in the paper is the Green function of the diffusion equation. To solve the corresponding boundary-value problem the Laplace transform with respect to the nonlocal parameter and the Hankel transform with respect to the radial coordinate are used. The Laplace transform is inverted analytically; inverting the Hankel transform the oscillatory integrals containing products of Bessel functions have been changed into integrands which decay exponentially, thus producing a solution more amenable to numerical quadrature. All classical singularities for stresses are eliminated.     

On nonlocal residuals and fluctuations

The nonlocal residual is a novel physical quantity introduced in the nonlocal field theory of mechanics. In this paper, the nonlocal residual and some related problems are discussed. Firstly, a representative theorem of nonlocal residual is proved, in which the relation between the nonlocal residual and the spatial distributed fluctuation of the interaction among microstructures in materials is established. The existence of nonlocal residuals of body force, body moment and energy is investigated in detail based on the objectivity of the balance equations. To meet the requirements in physics, an eigen-scale parameter is introduced into the nonlocal kernel. And the properties of nonlocal kernel are then discussed. Finally, the nonlocal hyperelastic constitutive equation is deduced through the representation of the nonlocal residual of energy. Results show that the nonlocality of hyperelastic constitutive equation comes directly from the interaction potential among microstructures within materials.     

Formulations of nonlocal continuum mechanics based on a new definition of stress tensor

Summary In this paper, attention is given to some basic problems in nonlocal continuum mechanics. Firstly, a surface-induced traction is introduced through nonlocal residuals of the surface. By it, a new measure of stress called the nonlocal stress is defined. The corresponding nonlocal stress formula and nonlocal balance equations distinguishing from the existing forms are established systematically. The existence of nonlocal residuals of body force, body moment and energy is investigated in detail based on the objectivity of the nonlocal balance equation of energy. Based on these results, the nonlocal hyperelastic constitutive equation is consistently deduced. Finally, the linear nonlocal elasticity is used to analyse the lattice contraction on a crystal surface induced by the surface-induced traction. The effects of the ``boundary layer' are discussed. Some interesting results are given.     

Longitudinal waves in a nonlocal rod by fractional Laplacian

ABSTRACT

The article deals with the longitudinal waves in a nonlocal elastic rod. Regarding the nonlocal elasticity the Eringeen model has been assumed; the novelty is that this model is described in terms of the fractional Laplace operator. The standing waves are obtained by numerical solutions of the fractional differential equation in one-dimensional continuum. The obtained results are in accordance with the ones reported in the literature and highlight the dispersion phenomenon. The effects of the nonlocal contribution and of the fractional Laplacian order are also analyzed.     

Differential quadrature method for nonlocal nonlinear vibration analysis of a boron nitride nanotube using sinusoidal shear deformation theory

This article presents a nonlocal sinusoidal shear deformation beam theory (SDBT) for the nonlinear vibration of single-walled boron nitride nanotubes (SWBNNTs). The surrounding elastic medium is simulated based on nonlinear Pasternak foundation. Based on the nonlocal differential constitutive relations of Eringen, the equations of motion of the SWBNNTs are derived using Hamilton's principle. Differential quadrature method (DQM) for the nonlinear frequency is presented, and the obtained results are compared with those predicted by the nonlocal Timoshenko beam theory (TBT). The effects of nonlocal parameter, vibrational modes, length, and elastic medium on the nonlinear frequency of SWBNNTs are considered.     

Vibration analysis of piezoelectrically actuated curved nanosize FG beams via a nonlocal strain-electric field gradient theory

In this article, nonlocal free vibration analysis of curved functionally graded piezoelectric (FGP) nanobeams is conducted using a Navier-type solution method. The model contains a nonlocal stress field parameter and also a nonlocal strain-electric field gradient parameter to capture the size effects. Inclusion of these nonlocal parameters introduces both stiffness-softening and stiffness-hardening effects in the analysis of curved nanobeams. Nonlocal governing equations of curved FGP nanobeam are obtained from Hamilton's principle based on the Euler–Bernoulli beam model. The results are validated with those of curved FG nanobeams available in the literature. Finally, the influences of electric voltage, length scale parameter, nonlocal parameter, opening angle, material composition, and slenderness ratio on vibrational characteristics of nanosize curved FG piezoelectric beams are explored. These results may be useful in accurate analysis and design of smart nanostructures constructed from piezoelectric materials.     

A review on the application of nonlocal elastic models in modeling of carbon nanotubes and graphenes

This paper reviews recent research studies on the application of the nonlocal continuum theory in modeling of carbon nanotubes and graphene sheets. A variety of nonlocal continuum models in modeling of the two materials under static and dynamic loadings are introduced and reviewed. The superiority of nonlocal continuum models to their local counterparts, the necessity of the calibration of the small-scale parameter, and the applicability of nonlocal continuum models are discussed. A brief introduction of the nonlocal beam, plate, and shell models is particularly presented. Summary and recommendations for future research are also provided. This paper is intended to provide an introduction of the development of the nonlocal continuum theory in modeling the two nano-materials, review the different nonlocal continuum models, and inspire further applications of the nonlocal continuum theory to nano-material modeling.     

The thermal effect on vibration of zigzag single walled carbon nanotubes using nonlocal Timoshenko beam theory

Based on nonlocal theory of thermal elasticity mechanics, a nonlocal elastic Timoshenko beam model is developed for free vibration analysis of zigzag single-walled carbon nanotube (SWCNT) considering thermal effect. The nonlocal constitutive equations of Eringen are used in the formulations. The equivalent Young’s modulus and shear modulus for zigzag SWCNT are derived using an energy-equivalent model. Results indicate significant dependence of natural frequencies on the temperature change as well as the chirality of zigzag carbon nanotube. These findings are important in mechanical design considerations of devices that use carbon nanotubes.     

A unified nonlocal formulation for bending,buckling and free vibration analysis of nanobeams

Abstract

A unified nonlocal formulation is developed for the bending, buckling, and vibration analysis of nanobeams. Theoretical formulations of eighteen nonlocal beam theories are presented by using unified formulation. Small scale effect is considered based on the nonlocal differential constitutive relations of Eringen. The governing equations of motion and associated boundary conditions of the nanobeam are derived using Hamilton's principle. Closed form solutions are presented for a simply supported boundary condition using Navier's solution technique. Numerical results for axial and transverse shear stress are first time presented in this study which will serve as a benchmark for the future research.     

Application of nonlocal beam models to double-walled carbon nanotubes under a moving nanoparticle. Part II: parametric study

The capabilities of the proposed nonlocal beam models in the companion paper in capturing the critical velocity of a moving nanoparticle as well as the dynamic response of double-walled carbon nanotubes (DWCNTs) under a moving nanoparticle are scrutinized in some detail. The role of the small-scale effect parameter, slenderness of DWCNTs and velocity of the moving nanoparticle on dynamic deflections and nonlocal bending moments of the innermost and outermost tubes as well as their maximum values are then investigated. The results reveal that the critical velocity increases with the slenderness of DWCNTs and the magnitude of the van der Waals interaction force. Nevertheless, the critical velocity generally decreases with the small-scale effect as well as the ratio of the mean diameter to the thickness of the innermost tube. Additionally, the predicted maximum dynamic deflections and nonlocal bending moments of the innermost and outermost tubes by using the nonlocal Euler?CBernoulli and Timoshenko beam theories are generally the lower and upper bounds of those obtained by the nonlocal higher-order beam theory (NHOBT). In the case of ??₁?<?20, the use of the NHOBT is highly recommended for more realistic prediction of dynamic response of DWCNTs under a moving nanoparticle.     

Nonlocal nonlinear formulations for bending of classical and shear deformation theories of beams and plates

The classical and shear deformation beam and plate theories are reformulated using the nonlocal differential constitutive relations of Eringen and the von Kármán nonlinear strains. The equations of equilibrium of the nonlocal beam theories are derived, and virtual work statements in terms of the generalized displacements are presented for use with the finite element model development. The governing equilibrium equations of the classical and first-order shear deformation theories of plates with the von Kármán nonlinearity are also formulated. The theoretical development presented herein should serve to obtain the finite element results and determine the effect of the geometric nonlinearity and nonlocal constitutive relations on bending response.     

Nonlocally related PDE systems for one-dimensional nonlinear elastodynamics

Complete dynamical PDE systems of one-dimensional nonlinear elasticity satisfying the principle of material frame indifference are derived in Eulerian and Lagrangian formulations. These systems are considered within the framework of equivalent nonlocally related PDE systems. Consequently, a direct relation between the Euler and Lagrange systems is obtained. Moreover, other equivalent PDE systems nonlocally related to both of these familiar systems are obtained. Point symmetries of three of these nonlocally related PDE systems of nonlinear elasticity are classified with respect to constitutive and loading functions. Consequently, new symmetries are computed that are: nonlocal for the Euler system and local for the Lagrange system; local for the Euler system and nonlocal for the Lagrange system; nonlocal for both the Euler and Lagrange systems. For realistic constitutive functions and boundary conditions, new dynamical solutions are constructed for the Euler system that only arise as symmetry reductions from invariance under nonlocal symmetries.