On the truth of nanoscale for nanobeams based on nonlocal elastic stress field theory： equilibrium, governing equation and static deflection

This paper has successfully addressed three critical but overlooked issues in nonlocal elastic stress field theory for nanobeams： （i） why does the presence of increasing nonlocal effects induce reduced nanostructural stiffness in many, but not consistently for all, cases of study, i.e., increasing static deflection, decreasing natural frequency and decreasing buckling load, although physical intuition according to the nonlocal elasticity field theory first established by Eringen tells otherwise？ （ii） the intriguing conclusion that nanoscale effects are missing in the solutions in many exemplary cases of study, e.g., bending deflection of a cantilever nanobeam with a point load at its tip; and （iii） the non-existence of additional higher-order boundary conditions for a higher-order governing differential equation. Applying the nonlocal elasticity field theory in nanomechanics and an exact variational principal approach, we derive the new equilibrium conditions, do- main governing differential equation and boundary conditions for bending of nanobeams. These equations and conditions involve essential higher-order differential terms which are opposite in sign with respect to the previously studies in the statics and dynamics of nonlocal nano-structures. The difference in higher-order terms results in reverse trends of nanoscale effects with respect to the conclusion of this paper. Effectively, this paper reports new equilibrium conditions, governing differential equation and boundary condi- tions and the true basic static responses for bending of nanobeams. It is also concluded that the widely accepted equilibrium conditions of nonlocal nanostructures are in fact not in equilibrium, but they can be made perfect should the nonlocal bending moment be replaced by an effective nonlocal bending moment. These conclusions are substantiated, in a general sense, by other approaches in nanostructural models such as strain gradient theory, modified couple stress models and experiments.     

Variational principles and conservation laws to the Burridge–Knopoff equation

We generate conservation laws for the Burridge–Knopoff equation which model nonlinear dynamics of earthquake faults by a new conservation theorem proposed recently by Ibragimov. One can employ this new general theorem for every differential equation (or systems) and derive new local and nonlocal conservation laws. Nonlocal conservation laws comprise nonlocal variables defined by the adjoint equations to the Burridge–Knopoff equation.     

Triply periodical particulate matrix compositesinvarying external stress fields

We consider a linear elastic composite medium, which consists of ahomogeneousmatrix containing aligned ellipsoidal uncoated or coated inclusions arranged in aperiodic arrayand subjected to inhomogeneous boundary conditions. The hypothesis of effectivefieldhomogeneity near the inclusions is used. The general integral equation obtained reducestheanalysis of infinite number of inclusion problems to the analysis of a finite number of inclusionsinsome representative volume element (RVE) . The integral equation is solved by theFouriertransform method as well as by the iteration method of the Neumann series ( first-orderapproximation) . The nonlocal macroscopic constitutive equation relating the unit cellaverages ofstress and strain is derived in explicit closed forms either of a differential equation ofasecond-order or of an integral equation. The employed of explicit relations fornumericalestimations of tensors describing the local and nonlocal effective elastic properties aswell asaverage stresses in the composites containing simple cubic lattices of rigid inclusions andvoids areconsidered.     

Galerkin Projections and Finite Elements for Fractional Order Derivatives

Ordinary differential equations (ODEs) with fractional order derivatives are infinite dimensional systems and nonlocal in time: the history of the state variable is needed to calculate the instantaneous rate of change. This nonlocal nature leads to expensive long-time computations (O(t ²) computations for solution up to time t). A finite dimensional approximation of the fractional order derivative can alleviate this problem. We present one such approximation using a Galerkin projection. The original infinite dimensional system is replaced with an equivalent infinite dimensional system involving a partial differential equation (PDE). The Galerkin projection reduces the PDE to a finite system of ODEs. These ODEs can be solved cheaply (O(t) computations). The shape functions used for the Galerkin projection are important, and given attention. The approximation obtained is specific to the fractional order of the derivative; but can be used in any system with a derivative of that order. Calculations with both global shape functions as well as finite elements are presented. The discretization strategy is improved in a few steps until, finally, very good performance is obtained over a user-specifiable frequency range (not including zero). In particular, numerical examples are presented showing good performance for frequencies varying over more than 7 orders of magnitude. For any discretization held fixed, however, errors will be significant at sufficiently low or high frequencies. We discuss why such asymptotics may not significantly impact the engineering utility of the method.     

Torsional static and vibration analysis of functionally graded nanotube with bi-Helmholtz kernel based stress-driven nonlocal integral model

A torsional static and free vibration analysis of the functionally graded nanotube(FGNT) composed of two materials varying continuously according to the power-law along the radial direction is performed using the bi-Helmholtz kernel based stress-driven nonlocal integral model. The differential governing equation and boundary conditions are deduced on the basis of Hamilton’s principle, and the constitutive relationship is expressed as an integral equation with the bi-Helmholtz kernel. Several nom...     

On a nonlocal stationary free-boundary problem arising in the confinement of a plasma in a Stellarator geometry

We prove the existence and some qualitative properties of the solution to a two-dimensional free-boundary problem modeling the magnetic confinement of a plasma in a Stellarator configuration. The nonlinear elliptic partial differential equation on the plasma region was obtained from the three-dimensional magnetohydrodynamic system by Hender & Carreras in 1984 by using averaging arguments and a suitable system of coordinates (Boozer's vacuum coordinates). The free boundary represents the separation between the plasma and vacuum regions, and the model is described by an inverse-type problem (some nonlinear terms of the equation are unknown). Using the zero net current condition for the Stellarator configurations, we reformulate the problem with the help of the notion of relative rearrangement, leading to a new problem involving nonlocal terms in the equation. We use an iterative algorithm and establish some new properties on the relative rearrangement in order to prove the convergence of the algorithm and then the existence of a solution.     

Reduced‐order modeling for nonlocal diffusion problems

In several settings, diffusive behavior is observed to not follow the rate of spread predicted by parabolic partial differential equations (PDEs) such as the heat equation. Such behaviors, often referred to as anomalous diffusion, can be modeled using nonlocal equations for which points at a finite distance apart can interact. An example of such models is provided by fractional derivative equations. Because of the nonlocal interactions, discretized nonlocal systems have less sparsity, often significantly less, compared with corresponding discretized PDE systems. As such, the need for reduced‐order surrogates that can be used to cheaply determine approximate solutions is much more acute for nonlocal models compared with that for PDEs. In this paper, we consider the construction, application, and testing of proper orthogonal decomposition (POD) reduced models for an integral equation model for nonlocal diffusion. For certain modeling parameters, the model we consider allows for discontinuous solutions and includes fractional Laplacian kernels as a special case. Preliminary computational results illustrate the potential of using POD to obtain accurate approximations of solutions of nonlocal diffusion equations at much lower costs compared with, for example, standard finite element methods. Copyright © 2016 John Wiley & Sons, Ltd.     

Bending of Euler–Bernoulli nanobeams based on the strain-driven and stress-driven nonlocal integral models: a numerical approach

Eringen’s nonlocal elasticity theory is extensively employed for the analysis of nanostructures because it is able to capture nanoscale effects. Previous studies have revealed that using the differential form of the strain-driven version of this theory leads to paradoxical results in some cases, such as bending analysis of cantilevers, and recourse must be made to the integral version. In this article, a novel numerical approach is developed for the bending analysis of Euler–Bernoulli nanobeams in the context of strain- and stress-driven integral nonlocal models. This numerical approach is proposed for the direct solution to bypass the difficulties related to converting the integral governing equation into a differential equation. First, the governing equation is derived based on both strain-driven and stress-driven nonlocal models by means of the minimum total potential energy. Also, in each case, the governing equation is obtained in both strong and weak forms. To solve numerically the derived equations, matrix differential and integral operators are constructed based upon the finite difference technique and trapezoidal integration rule. It is shown that the proposed numerical approach can be efficiently applied to the strain-driven nonlocal model with the aim of resolving the mentioned paradoxes. Also, it is able to solve the problem based on the strain-driven model without inconsistencies of the application of this model that are reported in the literature.     

Nonlocal vibration analysis of circular double-layered graphene sheets resting on an elastic foundation subjected to thermal loading

Based on the nonlocal elasticity theory, the vibra-tion behavior of circular double-layered graphene sheets (DLGSs) resting on the Winkler- and Pasternak-type elas-tic foundations in a thermal environment is investigated. The governing equation is derived on the basis of Eringen’s nonlocal elasticity and the classical plate theory (CLPT). The initial thermal loading is assumed to be due to a uniform temperature rise throughout the thickness direc-tion. Using the generalized differential quadrature (GDQ) method and periodic differential operators in radial and cir-cumferential directions, respectively, the governing equation is discretized. DLGSs with clamped and simply-supported boundary conditions are studied and the influence of van der Waals (vdW) interaction forces is taken into account. In the numerical results, the effects of various parameters such as elastic medium coefficients, radius-to-thickness ratio, thermal loading and nonlocal parameter are examined on both in-phase and anti-phase natural frequencies. The results show that the thermal load and elastic foundation respec-tively decreases and increases the fundamental frequencies of DLGSs.     

Nonlocal implicit gradient-enhanced elasto-plasticity for the modelling of softening behaviour

An improved gradient-enhanced approach for softening elasto-plasticity is proposed, which in essence is fully nonlocal, i.e. an equivalent integral nonlocal format exists. The method utilises a nonlocal field variable in its constitutive framework, but in contrast to the integral models computes this nonlocal field with a gradient formulation. This formulation is considered ‘implicit’ in the sense that it strictly incorporates the higher-order gradients of the local field variable indirectly, unlike the common (explicit) gradient approaches. Furthermore, this implicit gradient formulation constitutes an additional partial differential equation (PDE) of the Helmholtz type, which is solved in a coupled fashion with the standard equilibrium condition. Such an approach is particularly advantageous since it combines the long-range interactions of an integral (nonlocal) model with the computational efficiency of a gradient formulation. Although these implicit gradient approaches have been successfully applied within damage mechanics, e.g. for quasi-brittle materials, the first attempts were deficient for plasticity. On the basis of a thorough comparison of the gradient-enhancements for plasticity and damage this paper rephrases the problem, which leads to a formulation that overcomes most reported problems. The two-dimensional finite element implementation for geometrically linear plain strain problems is presented. One- and two-dimensional numerical examples demonstrate the ability of this method to numerically model irreversible deformations, accompanied by the intense localisation of deformation and softening up to complete failure.     

Buckling analysis of functionally graded nanobeams under non-uniform temperature using stress-driven nonlocal elasticity

In this work, the size-dependent buckling of functionally graded(FG)Bernoulli-Euler beams under non-uniform temperature is analyzed based on the stressdriven nonlocal elasticity and nonlocal heat conduction. By utilizing the variational principle of virtual work, the governing equations and the associated standard boundary conditions are systematically extracted, and the thermal effect, equivalent to the induced thermal load, is explicitly assessed by using the nonlocal heat conduction law. The ...     

Surface effects of adsorption-induced resonance analysis on micro/nanobeams via nonlocal elasticity

The governing differential equation of micro/nanbeams with atom/molecule adsorption is derived in the presence of surface effects using the nonlocal elasticity.The effects of the nonlocal parameter,the adsorption density,and the surface parameter on the resonant frequency of the micro/nanobeams are investigated.It is found that,in addition to the nonlocal parameter and the surface parameter,the bending rigidity and the adsorption-induced mass exhibit different behaviors with the increase in the adsorption density depending on the adatom category and the substrate material.     

Potential Symmetries and Associated Conservation Laws with Application to Wave Equations

It has been shown that one can generate a class of nontrivial conservation laws for second-order partial differential equations using some recent results dealing with the action of any Lie–Bäcklund symmetry generator of the equivalentfirst-order system on the respective conservation law. These conservedvectors are nonlocal as they are constructed from associatednonlocal symmetries of the partial differential equation. The method canbe successfully extended to association with genuine nonlocal(potential) symmetries. However, it usually involves solving moredifficult systems of partial differential equations which may not alwaysbe easy to uncouple.     

Analysis of the Volume-Constrained Peridynamic Navier Equation of Linear Elasticity

Well-posedness results for the state-based peridynamic nonlocal continuum model of solid mechanics are established with the help of a nonlocal vector calculus. The peridynamic strain energy density for an elastic constitutively linear anisotropic heterogeneous solid is expressed in terms of the field operators of that calculus, after which a variational principle for the equilibrium state is defined. The peridynamic Navier equilibrium equation is then derived as the first-order necessary conditions and are shown to reduce, for the case of homogeneous materials, to the classical Navier equation as the extent of nonlocal interactions vanishes. Then, for certain peridynamic constitutive relations, the peridynamic energy space is shown to be equivalent to the space of square-integrable functions; this result leads to well-posedness results for volume-constrained problems of both the Dirichlet and Neumann types. Using standard results, well-posedness is also established for the time-dependent peridynamic equation of motion.     

On invariance properties of an extended energy balance

Gradient plasticity theories are of utmost importance for accounting for size effects in metals, especially on the grain scale. Today, there are several methods used to derive the governing equations for the additional degrees of freedom in gradient plasticity theories. Here, the equivalence between an extended principle of virtual power and an extended energy balance is shown. The energy balance of a Boltzmann continuum is supplemented by contributions based on a scalar-valued degree of freedom. It is considered to be invariant with respect to a change of observer. This yields unambiguously the existence of a corresponding micro-stress vector, which is presumed from the outset in the context of an extended principle of virtual power. A thermodynamically consistent nonlocal evolution equation for the additional, scalar-valued degree of freedom is obtained by evaluation of the dissipation inequality in terms of the Clausius–Duhem inequality. Partitioning the nonlocal flow rule yields a partial differential equation, often referred to as micro-force balance. The approach presented is applied to derive a slip gradient crystal plasticity theory regarding single slip. Finally, the distribution of the plastic slip is exemplified with respect to a laminate material consisting of an elastic and an elastoplastic phase.     

Nonlocal stress field of interface dislocations

Summary The basic theory of nonlocal elasticity is stated with emphasis on the difference between the nonlocal theory and classical continuum mechanics. The concept of Nonlocal Interface Residual (NIR) is introduced in nonlocal theory. With the concept of NIR and the nonlocal constitutive equation, we calculate nonlocal stresses due to an edge dislocation on the interface of bi-materials. The nonlocal stress distribution along an interface is quite different from the classical one. Instead of the singularity in the dislocation core, nonlocal stress gives a finite value in the core. A maximum of the stress is also found near the dislocation core. Received 27 May 1997; accepted for publication 1 July 1997     

On an Inviscid Model for Incompressible Two-Phase Flows with Nonlocal Interaction

We consider a diffuse interface model which describes the motion of an ideal incompressible mixture of two immiscible fluids with nonlocal interaction in two-dimensional bounded domains. This model consists of the Euler equation coupled with a convective nonlocal Cahn-Hilliard equation. We establish the existence of globally defined weak solutions as well as well-posedness results for strong/classical solutions.     

NEW POINTS OF VIEW ON THE NONLOCAL FIELD THEORY AND THEIR APPLICATIONS TO THE FRACTURE MECHANICS(Ⅲ)——RE-DISCUSS THE LINEAR THEORY OF NONLOCAL ELASTICITY

In this paper, it is proven that the balance equation of energy is the first integral of the balance equation of momentum in the linear theory of nonlocal elasticity. In other words, the balance equation of energy is not an independent one. It is also proven that the residual of nonlocal body force identically equals zero. This makes the transform formula of the nonlocal residual of energy much simpler. The linear nonlocal consitutive equations of elastic bodies are deduced in details, and a new formula to calculate the antisymmetric stress is given. Foundation item: the Natural Science Foundation of Jiangsu Province, China (BK97063)