Nonlocal Solutions Research Articles

The investigation of the nonlocal longitudinal stress waves with modified couple stress theory

In the present work, the propagation of longitudinal stress waves is investigated using a modified couple stress theory. The analysis of wave motion is based on a Love rod model including the effects of lateral deformation. The present analysis also considers the effect of shear stress components. By applying Hamilton’s principle, the explicit nonlocal elasticity solution is obtained, and the effects of shear stress and length scale parameter are discussed.

Exact nonlocal solution for postbuckling of single-walled carbon nanotubes

The purpose of this paper is to present exact and efficient analytical expressions for the postbuckling configurations of single-walled carbon nanotubes with different boundary conditions. The nonlinear governing partial-integral differential equations are derived based on Eringen's nonlocal elasticity model and the Euler–Bernoulli beam theory. The geometric nonlinearity is taken into account, which arises from the mid-plane stretching. The exact nonlocal model results can be conveniently used to assess the sensitivity of the small-scale parameter on the nanotubes postbuckling load–deflection relationship. The accuracy of the solution is demonstrated by comparing the critical buckling load results with those available in literature. The influences of small-scale parameter, various end conditions as well as nonlinearity on the postbuckling deformation are examined.

Localization of the SFT inspired nonlocal linear models and exact solutions

A general class of gravitational models driven by a nonlocal scalar field with a linear or quadratic potential is considered. We study the action with an arbitrary analytic function $F(\Box)$, which has both simple and double roots. The way of localization of nonlocal Einstein equations is generalized on models with linear potentials. Exact solutions in the Friedmann-Robertson-Walker and Bianchi I metrics are presented.

Non-local separable solutions of two interacting particles in a harmonic trap

We calculate the energy levels of two particles trapped in a harmonic potential. The actual two-body potential, assumed to be spherically symmetric, is replaced by a projective operator (non-local separable potential) to determine the energy levels in a closed form. This approach overcomes the limitations of the regularized Fermi pseudopotential when the characteristic length of the two-body interaction potential is of the order of the size of the harmonic trap. In addition, we recover the results obtained with the Fermi pseudopotential when the length of the interaction is much smaller than the size of the trap.

Non-local theory solution of two collinear mode-I permeable cracks in a magnetoelectroelastic composite material plane

The non-local theory solution of two collinear mode-I permeable cracks in a magnetoelectroelastic composite material plane was investigated using the generalized Almansi's theorem and the Schmidt method. The problem was formulated through Fourier transform into two pairs of dual integral equations, in which the unknown variables are the jumps in displacements across the crack surfaces. To solve the dual integral equations, the displacement jumps across the crack surfaces were directly expanded as a series of Jacobi polynomials. Numerical examples were provided to show the effects of crack length, the distance between two collinear cracks and the lattice parameter on the stress field, the electric displacement field and the magnetic flux field near the crack tips. Unlike the classical elasticity solutions, it is found that no stress, electric displacement or magnetic flux singularities are present at the crack tips in a magnetoelectroelastic composite material plane. The non-local elastic solutions yield a finite hoop stress at the crack tip, thus allowing us to use the maximum stress as a fracture criterion.

Non-local theory solution of a mode-I crack in a piezoelectric/piezomagnetic composite material plane

The non-local theory solution of a mode-I permeable crack in a piezoelectric/piezomagnetic composite material plane was given by using the generalized Almansi’s theorem and the Schmidt method in this paper. The problem was formulated through Fourier transform into two pairs of dual integral equations, in which the unknown variables are the displacement jumps across the crack surfaces. To solve the dual integral equations, the displacement jumps across the crack surfaces were directly expanded as a series of Jacobi polynomials. Numerical examples were provided to show the effects of the crack length and the lattice parameter on the stress field, the electric displacement field and the magnetic flux field near the crack tips. Unlike the classical elasticity solutions, it is found that no stress, electric displacement and magnetic flux singularities are present at the crack tips in piezoelectric/piezomagnetic composite materials. The non-local elastic solution yields a finite hoop stress at the crack tip, thus allowing us to use the maximum stress as a fracture criterion.

Numerical fluid solutions for nonlocal electron transport in hot plasmas: Equivalent diffusion versus nonlocal source

Flux limitation and preheat are important processes in electron transport occurring in laser produced plasmas. The proper calculation of both of these has been a subject receiving much attention over the entire lifetime of the laser fusion project. Where nonlocal transport (instead of simple single flux limit) has been modeled, it has always been with what we denote the equivalent diffusion solution, namely treating the transport as only a diffusion process. We introduce here a new approach called the nonlocal source solution and show it is numerically viable for laser produced plasmas. It turns out that the equivalent diffusion solution generally underestimates preheat. Furthermore, the advance of the temperature front, and especially the preheat, can be held up by artificial 'thermal barriers'. The nonlocal source method of solution, on the other hand more accurately describes preheat and can stably calculate the solution for the temperature even if the heat flux is up the gradient.

A similarity solution in order to solve the governing equations of laminar separated fluids with a flat plate

In this paper, the development of local and non-local similarity solutions for laminar flow and heat transfer between two separated fluids is described. This paper focuses on the extension of similarity solutions of convective surface boundary condition. This new case represents two adjacent fluids separated by a flat plate, and moving parallel to each other, where the convective heat transfer coefficient of the fluid heating the plate on its lower surface is proportional to x - 1 / 2 . Numerical and analytical solutions are available to solve this boundary value problem. Within the first case, shooting method is applied; furthermore, Runge–Kutta fourth order method is used for integration over the whole boundary layer. Numerical solutions of the resulting similarity energy equation are represented for various Prandtl numbers and a range of values of the parameter characterizing the hot fluid convection process. In addition, analytical exact series solutions are provided for all different Prandtl numbers, although for cold fluids with low Prandtl numbers, a compact solution is also obtained. Finally, an appropriate range of Prandtl number is obtained in which compact and exact solution have a good agreement.

Dynamic Instability of Nanorods/Nanotubes Subjected to an End Follower Force

This paper presents an investigation on the dynamic instability of cantilevered nanorods/nanotubes subjected to an end follower force. Eringen's nonlocal elasticity theory is employed to allow for the small length scale effect in the considered dynamic instability problem. The general solution for the governing differential equation is obtained and the dynamic instability characteristic equation is derived by applying the boundary conditions. Exact critical load factors are obtained. These nonlocal solutions are compared with the classical local solutions to assess the sensitivity of the small length scale effect on the critical load factors and flutter mode shapes. It is found that the small length scale effect decreases the critical load and the corresponding frequency parameters as well as reduces the severity of the double-curvature flutter mode shape.

Comment on “New ansatz for metric operator calculation in pseudo-Hermitian field theory”

In a recent Brief Report by Shalaby, a new first-order perturbative calculation of the metric operator for an $i{\ensuremath{\phi}}^{3}$ scalar field theory is given. It is claimed that the incorporation of derivative terms in the ansatz for the metric operator results in a local solution, in contrast to the nonlocal solution previously obtained by Bender, Brody, and Jones. Unfortunately, Shalaby's calculation is not valid because of sign errors.

ON THE SIGNIFICANCE OF THE WEYL CURVATURE IN A RELATIVISTIC COSMOLOGICAL MODEL

The Weyl curvature includes the Newtonian field and an additional field, the so-called anti-Newtonian. In this paper, we use the Bianchi and Ricci identities to provide a set of constraints and propagations for the Weyl fields. The temporal evolutions of propagations manifest explicit solutions of gravitational waves. We see that models with purely Newtonian field are inconsistent with relativistic models and obstruct sounding solutions. Therefore, both fields are necessary for the nonlocal nature and radiative solutions of gravitation.

Vibration analysis of nanoplates under uniaxial prestressed conditions via nonlocal elasticity

In this article, nonlocal elasticity theory is applied to investigate the vibration response of nanoplates under uniaxially prestressed conditions. Nonlocal elasticity theory takes into account the small-size effects when dealing with nanostructures. Nonlocal governing equations of the prestressed nanoplate are derived and presented. Differential quadrature method is being utilized and numerical frequency solutions are obtained. Influence of small scale and uniaxial preload on the nonlocal frequency solutions is investigated. It is observed that the frequencies for nanoplates under uniaxially prestressed conditions employing classical plate theory are overestimated compared to nonlocal plate solutions. Considering the nonlocal effects, smaller critical compressive load is required to reach the buckling state of a flexural mode compared to the classical plate theory. The present research work thus reveals that the nonlocal parameter, aspect ratios, boundary conditions, and initial uniaxial prestress have significant effects on vibration response of the nanoplates.

Optimal gait and form for animal locomotion

We present a fully automatic method for generating gaits and morphologies for legged animal locomotion. Given a specific animal's shape we can determine an efficient gait with which it can move. Similarly, we can also adapt the animal's morphology to be optimal for a specific locomotion task. We show that determining such gaits is possible without the need to specify a good initial motion, and without manually restricting the allowed gaits of each animal. Our approach is based on a hybrid optimization method which combines an efficient derivative-aware spacetime constraints optimization with a derivative-free approach able to find non-local solutions in high-dimensional discontinuous spaces. We demonstrate the effectiveness of this approach by synthesizing dynamic locomotions of bipeds, a quadruped, and an imaginary five-legged creature.

Small-scale effect on the vibration of nonuniform nanocantilever based on nonlocal elasticity theory

In this article, vibration response of nanocantilever is investigated considering nonuniformity in the cross sections. Using nonlocal elasticity theory, governing differential equations are established. Differential quadrature (DQ) method is being employed and natural frequencies of the structure are obtained. The present study shows that the nonlocal frequency solutions of nanocantilever are larger compared to the classical (local) solutions till a critical height ratio (CHR). Beyond the CHR, nonlocal solutions are lower than the classical (local) solutions.

The nonlocal theory solution of a Mode-I crack in functionally graded materials

The behavior of a Mode-I finite crack in functionally graded materials is investigated using the non-local theory. To make the analysis tractable, it is assumed that the shear modulus varies exponentially with coordinate vertical to the crack. The problem in this paper can be solved through the Fourier transform with the help of two pairs of dual integral equations, in which the unknown variables are jumps of displacements across crack surfaces. To solve dual integral equations, the jumps of displacements across crack surfaces are directly expanded in a series of Jacobi polynomials. Unlike the classical elasticity solutions, it is found that no stress singularities are present at crack tips. The non-local elastic solutions yield a finite stress at crack tips, thus allowing us to use the maximum stress as a fracture criterion. Numerical examples are provided to show the effects of the crack length, the parameter describing the functionally graded materials, the lattice parameter of materials and the materials constants upon the stress fields near crack tips.

The Non-Local Theory Solution of Orthotropic Composite Materials on the Stress Field near the Crack Tips

The Non-Local Theory Solution of Orthotropic Composite Materials on the Stress Field near the Crack Tips

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

This paper is concerned with the use of the nonlocal Timoshenko beam model for free vibration analysis of single-walled carbon nanotubes (CNTs) including the thermal effect. Unlike the Euler beam model, the Timoshenko beam model allows for the effects of transverse shear deformation and rotary inertia. These effects become significant for CNTs with small length-to-diameter ratios that are normally encountered in applications such as nanoprobes. The elastic Timoshenko beam model is reformulated using the nonlocal differential constitutive relations of Eringen (1972 Int. J. Eng. Sci. 10 1–16). The study focuses on the wave dispersion caused not only by the rotary inertia and the shear deformation in the traditional Timoshenko beam model but also by the nonlocal elasticity characterizing the microstructure of CNTs in a wide frequency range up to terahertz. Numerical results are presented using the nonlocal beam theory to bring out the effect of both the nonlocal parameter and the temperature change on the properties of transverse vibrations of CNTs. The exact nonlocal Timoshenko beam solution presented here should be useful to engineers who are designing microelectromechanical and nanoelectromechanical devices.

Nonlocal instantons and solitons in string models

We study a class of nonlocal systems which can be described by a local scalar field diffusing in an auxiliary radial dimension. As examples p-adic, open and boundary string field theory are considered on Minkowski, Friedmann–Robertson–Walker and Euclidean metric backgrounds. Starting from distribution-like initial field configurations which are constant almost everywhere, we construct exact and approximate nonlocal solutions. The Euclidean p-adic lump is interpreted as a solitonic brane, and the Euclidean kink of supersymmetric open string field theory as an instanton. Some relations between solutions of different string theories are highlighted also thanks to a reformulation of nonlocal systems as fixed points in a renormalization group flow.

Near-dissipation range in nonlocal turbulence

Viscous perturbations to scaling are studied in the near-dissipation range of isotropic turbulence. A quantitative relationship between effective strain of the nonlocal interactions and viscosity has been found. It is shown that nonlocal interactions determine the energy spectrum in isotropic turbulence at small Reynolds numbers. It is also shown that for moderate Reynolds numbers the bottleneck effect is determined by the same nonlocal interactions. The role of the large- and small-scale covariances at the nonlocal interactions and in energy balance has been investigated. A possible hydrodynamic mechanism of the nonlocal solution instability at large scales has been briefly discussed. Analogous approach has also been developed for passive scalar and for the energy dissipation rate spectrum. All results are supported by a comparison with the data of the laboratory experiments and numerical simulations.

Cracking Cohesive Law Thermodynamically Equivalent to a Non-Local Damage Model

This paper deals with the transition from a localized damage state to crack formation. Several attempts have already been made in this field. Our approach is in the continuity of studies where thermodynamic considerations lead to the definition of an equivalent crack concept. The main idea consists in replacing a damaged localized zone by a crack in order to recover the same amount of dissipated energy. On the one hand, a nonlocal model is used to modelize accurately localized damage. On the other hand, an elastic model which authorizes the formation of a crack described by a cohesive zone model is used. This cohesive zone model is defined thermodynamically in order to be in concordance with the damage model. The method allows obtaining the cohesive zone model traction curve from the knowledge of the nonlocal damage model solution. The numerical implementation is done using a Lagrangian multiplier that ensures the energetic equivalence between both models.