General Nonlocal Theory Research Articles

Can quantum theory be underpinned by a non-local hidden variable theory?

In this paper we consider the description by a general Bell-type non-local hidden variable theory of bipartite quantum states with two observables per sub-system. We derive Bell inequalities of the Collins-Gisin.-Liden-Massar-Popescu type which involve combinations of the probabilities of related outcomes for measurements for the four pairs of sub-system observables. It is shown that the corresponding quantum theory expressions violate the Bell inequalities in the case of the maximally entangled state of the bipartitite system. The CHSH Bell inequality is also derived from this general CGLMP Bell-type non-local hidden variable theory. This shows that quantum theory can not be underpinned by a Bell-type non-local hidden variable theory. So as a general Bell-type local hidden variable theory has already been shown to conflict with quantum theory, it follows that quantum theory can not be understood in terms of any CGLMP Bell-type hidden variable theory—local or non-local.

The Euclidean $\phi^{4}_{2}$ theory as a limit of an interacting Bose gas

We prove that the complex Euclidean field theory with local quartic self-interaction in two dimensions arises as a limit of an interacting Bose gas at positive temperature, when the density of the gas becomes large and the range of the interaction becomes small. The field theory is supported on distributions of negative regularity, which requires a renormalization by divergent mass and energy counterterms. We obtain convergence of the relative partition function and uniform convergence of the renormalized reduced density matrices. The proof is based on three main ingredients: (a) a quantitative analysis of the infinite-dimensional saddle point argument for the functional integral introduced by Fröhlich et al. (2022) using continuity properties of Brownian paths, (b) a Nelson-type estimate for a general nonlocal field theory in two dimensions, and (c) repeated Gaussian integration by parts in field space to obtain uniform control on the renormalized correlation functions. As a byproduct of our proof, in two and three dimensions we also extend the results on the mean-field limit by Fröhlich et al. (2022) and Lewin et al. (2021) to unbounded interaction potentials satisfying the optimal integrability conditions proposed by Bourgain (1997).

Transversely heterogeneous nonlocal Timoshenko beam theory: A reduced-order modeling via distributed-order fractional operators

Nonlocal effects are widely identified in micro/nano systems and have been studied extensively in recent decades. Nevertheless, existing studies mostly focus on the modeling of homogeneous nonlocal systems and as a result, cannot model micro/nanostructures with heterogeneously distributed nonlocal effects. This study introduces a reduced-order nonlocal Timoshenko beam model that facilitates precise modeling of micro-elastic beams with layer-wise heterogeneous nonlocal effects. First, a fully-resolved two-dimensional (2D) plane-strain fractional-order nonlocal elastic framework is developed to provide reference solutions for 2D nonlocal beam problems. Building upon this general nonlocal elasticity theory, an equivalent one-dimensional (1D) nonlocal Timoshenko model is then developed. The layer-wise heterogeneous nonlocal information is captured by using distributed-order (DO) operators, and the impacts arising from the nonlocal heterogeneity are further characterized by introducing two auxiliary parameters. Both 1D and 2D approaches are applied to simulate the mechanical responses of nonlocal beams. Direct comparisons of numerical simulations produced by either the 1D or the fully-resolved 2D model confirm that the DO Timoshenko beam formulation (together with the two auxiliary parameters) can capture not only the overall beam deflection but also the additional shear effect induced by the heterogeneous nonlocality. Computational cost assessment also indicates the proposed approach’s superior performance. The proposed DO Timoshenko model enables model-order reduction without compromising the heterogeneous nonlocal description of the material hence leading to an efficient and accurate reduced-order nonlocal modeling approach that addresses the limitations of prior nonlocal microstructural theories and extends applicability to a broader range of heterogeneous nano/microstructures.

Applicability and efficacy of Galerkin-based approximation for solving the buckling and dynamics of nanobeams with higher-order boundary conditions

In this research effort, the general nonlocal elasticity theory is applied to an axially-loaded Euler-Bernoulli nanobeam with varying boundary conditions. In applying the general nonlocal elasticity theory, a modified version of Eringen's nonlocal elasticity theory, a sixth-order partial differential equation is obtained for the transverse governing equation of motion. Obtaining the sixth-order equation created the need for additional non-classical beam boundary conditions. These boundary conditions are determined using the weighted residual approach and are physically interpreted as higher-order force and moment terms. This study investigates the buckling and dynamic responses of nanobeams using a Galerkin-based approximation with different trial mode shapes, as it relates to the exact solution. As such, comparisons are made between using three distinct types guess functions in the approximation, namely, the exact unloaded classical mode shapes, exact unloaded Eringen nonlocal mode shapes, and exact unloaded general nonlocal mode shapes. Unloaded mode shapes refer to the mode shapes in which the applied axial load is neglected or set to zero. Additional studies are conducted focusing on direct comparison between Eringen nonlocal and general nonlocal beams for both equal and unequal nonlocal parameters. It is shown that selection of an appropriate guess function that satisfies all boundary conditions is necessary for the approximate solution. Additionally, the response of Eringen and general nonlocal nanobeams with equal nonlocal parameters and hinged-hinged boundary conditions is identical using both the exact and approximate solutions. Finally, it is demonstrated that the general nonlocal dynamic response follows a different trajectory than the Eringen nonlocal dynamic response as a function of the applied axial load. The findings of this investigation may be used by other researchers to simplify analysis procedures and to better understand the fundamental relationships existing in the size-dependent governing equations and boundary conditions for nanobeam-based systems.

Application of sensitivity analysis and uncertainty quantification methods on the dynamic response of general nonlocal beams

There is a gap to be filled in the literature regarding uncertainties that may exist in nanobeams’ material and geometric properties, such as the nonlocal parameters, Young's modulus, Poisson ratio, density, and beam's length, width, and height. Considering the fact there are so many input parameters that can affect the response of the system, parametric studies that only focus on single parameters may limit the usefulness of the obtained output. Thus, in this effort, local and global sensitivity analysis and uncertainty quantification techniques are applied to determine the most influential parameters on the acoustic dispersion curves and the linear response of nanobeams which are modeled utilizing Euler–Bernoulli beam theory and the general nonlocal theory. The sensitivity analysis and uncertainty quantification methods include expanded parametric studies around an ideal or baseline system configuration, the Morris method elementary effects, Sobol’ sensitivity indices, Pearson correlation coefficients, and output distribution curves. Various input probability distributions are used for the input parameters uncertainties and the effects of all parameters on the ideal configurations are investigated for both acoustic dispersions and first natural frequency. For this study, there is great agreement in input parameter ranking between the methods. The introduced methodologies can be used for many other complex dynamical systems to show the limits of applicability of each method and determine the most influential input parameters and dominant output configurations.

Analytical Solution for Buckling Behavior of FGM Plate Considering Surface Effect Based on General Third-Order Plate Theory and Non-local Theory

In this paper, the buckling characteristic of FGM plate considering the surface effect is studied based on general third-order plate theory and non-local theory. The surface effect of FGM plate is captured by the surface elasticity theory. The Kirchhoff hypothesis is released by employing parabolic variation of transverse shear strains. By using Navier solution technique, analytical solutions of buckling loads of FGM plate with surface effect are given, and detailed parametric studies are presented to show the relationship between surface effects and the plate thickness, power-law index, surface residual stress, surface moduli and non-local parameter. Furthermore, the surface effect on the buckling characteristic of FGM plate is also discussed.

Observations on the general nonlocal theory applied to axially loaded nanobeams

Static and dynamic analyses for the pre- and post-buckling behavior of an axially loaded nanobeam modeled with a general form of Eringen’s nonlocal theory are conducted and discussed. The general nonlocal theory is a modified form of Eringen’s nonlocal elasticity theory that, contrary to Eringen’s, considers separate attenuation functions to account for the long-range interactions for the two different material moduli and thus introduces an additional nonlocal parameter. The nanobeam is modeled considering Euler–Bernoulli beam theory and the von Karman geometric nonlinearity for midplane stretching in end-constrained beams. Three sets of boundary conditions are studied, namely clamped–clamped, clamped-hinged, and hinged–hinged. The governing equations are derived by virtue of the Hamilton’s principle. Introducing the general nonlocal theory into a beam system increases the order of the resultant transverse equation of motion from fourth to sixth, thus requiring two additional higher-order boundary conditions. These higher-order boundary conditions are derived using a weighted residual approach and are thus variationally consistent. Through this effort, a focus is placed on the effects that these higher-order boundary conditions have on the static and dynamic responses of the system. Specifically, the nonlocal parameters, Poisson ratio, boundary conditions, and axial load are varied. The static critical buckling loads, bifurcation diagrams, and static post-buckling configurations are examined and presented. Additionally, through a linear eigenvalue analysis, the natural frequencies and mode shapes in the pre- and post-buckling regimes are examined. It is shown that the system is very sensitive to the selected nonlocal parameters and higher-order boundary conditions for both the static and dynamic responses. It is the hope that this expanded model can be used by other researchers to accurately model nanobeams composed of materials outside the limits of applicability of Eringen’s nonlocal theory.

The vibration of two-dimensional imperfect functionally graded (2D-FG) porous rotating nanobeams based on general nonlocal theory

A comprehensive vibrational analysis of bi-directional functionally graded (2D-FG) rotating nanobeams with porosities is studied for the first time. The beam is modeled based on general nonlocal theory (GNT) where the beam governing equations are derived depending on two different nonlocal parameters. Unlike Eringen’s conventional form of nonlocal theory, the general nonlocal theory can reveal both hardening and softening behaviors of the material. Here, the attenuation functions are altered in both transverse and longitudinal directions of 2D-FG nanobeam. This feature, which has a significant effect on the vibrational characteristics, has not been considered in previous studies. Moreover, to estimate the effects of the higher-order transverse shear strains on the vibration of the nanobeam, Reddy’s beam theory (RBT), which includes higher-order shear deformation, is employed. The material properties of the 2D-FG rotating nanobeam vary both in the length and thickness directions according to a power law. The generalized differential quadrature method (GDQM) is used to predict the vibration response. Also, the effects of material variation along the length and thickness directions, the rotating velocity of the nanobeam, the porosity volume fraction and the length to thickness ratio of the rotating nanobeam are illustrated and discussed in detail. The investigations performed in this study expose new phenomena for the vibration of nanobeams.

On wave propagation in two-dimensional functionally graded porous rotating nano-beams using a general nonlocal higher-order beam model

This paper studies the wave propagation of two-dimensional functionally graded (2D-FG) porous rotating nano-beams for the first time. The rotating nano-beams are made of two different materials, and the material properties of the nano-beams alter both in the thickness and length directions. The general nonlocal theory (GNT) in conjunction with Reddy's beam model are employed to formulate the size-dependent model. The GNT efficiently models the dispersions of acoustic waves when two independent nonlocal fields are modelled for the longitudinal and transverse acoustic waves. The governing equations of motion for the 2D-FG porous rotating nano-beams are established using Hamilton's principle as a function of the axial force due to centrifugal stiffening and displacement. The analytic solution is applied to obtain the results and solve the governing equations. The effect of the features of different parameters such as functionally graded power indexes, porosity, angular velocity, and material variation on the wave propagation characteristics of the rotating nano-beams are discussed in detail.

A non-local fractional stress–strain gradient theory

A generalized non-local stress–strain gradient theory is presented using fractional calculus. The proposed theory includes as a special case: the classical theory; the non-local strain gradient theory; the Eringen non-local theory; the strain gradient theory; the general Eringen non-local theory; and the general strain gradient theory. This new formulation is therefore more comprehensive and more complete to model physical phenomena. Its application has been shown in free vibration, buckling and bending of simply supported (S–S) nano-beams. The non-linear governing equations have been solved by the Galerkin method. Furthermore the effects of different (additional) model parameters like: the length scale parameter; the non-local parameter; and different orders (integer and non-integer) of strain and stress gradients have been shown.

Influence of long-range interatomic and interlayer interactions on dispersion of acoustic waves by multilayer graphene

In this study, effects of the long-range interatomic and interlayer interactions on the dispersion of acoustic waves by multilayer graphene are investigated. A hybrid model that accounts for the long-range interatomic and interlayer interactions is developed. In the context of this model, the interatomic interactions are modeled based on the general nonlocal theory. The general nonlocal theory effectively models the dispersions of acoustic waves where it defines two independent nonlocal fields for the longitudinal and transverse acoustic waves. Interlayer interactions are defined to model the van der Waals interaction between the graphene layers. A new approach is proposed to determine the mechanical properties and the equivalent thickness of a single-layer graphene. The Young's modulus and Poisson's ratio of a single-layer graphene is obtained by ∼345MPa and ∼0.08, respectively. The equivalent thickness of a graphene sheet is determined by ∼0.057nm. An experimental validation for the dispersions of four-layer graphene is carried out where a good assent is observed. A parametric study is then presented to demonstrate the significant influence of the long-rang interatomic and interlayer interactions on the dispersion of incident acoustic waves by multilayer graphene.

Correction of local elasticity for nonlocal residuals: application to Euler–Bernoulli beams

Complications exist when solving the field equation in the nonlocal field. This has been attributed to the complexity of deriving explicit forms of the nonlocal boundary conditions. Thus, the paradoxes in the existing solutions of the nonlocal field equation have been revealed in recent studies. In the present study, a new methodology is proposed to easily determine the elastic nonlocal fields from their local counterparts without solving the field equation. This methodology depends on the iterative-nonlocal residual approach in which the sum of the nonlocal fields is treaded as a residual field. Thus, in this study the corrections of the local linear and nonlinear elastic fields for the nonlocal residuals in materials are presented. These corrections are formed based on the general nonlocal theory. In the context of the general nonlocal theory, two distinct nonlocal parameters are introduced to form the constitutive equations of isotropic elastic continua. In this study, it is demonstrated that the general nonlocal theory outperforms Eringen’s nonlocal theory in accounting for the impacts of the material’s Poisson’s ratio on its mechanics. To demonstrate the effectiveness of the proposed approach, the corrections of the local static bending, vibration, and buckling characteristics of Euler–Bernoulli beams are derived. Via these corrections, bending, vibration, and buckling behaviors of simple-supported nonlocal Euler–Bernoulli beams are determined without solving the beam’s equation of motion.

Poisson's ratio effects on the mechanics of auxetic nanobeams

Poisson ratio is an important mechanical property that reveals the deformation patterns of materials. A positive Poisson ratio is a feature of the majority of materials. Some materials, however, display auxetic behaviors (i.e. possess negative Poisson ratios). Indeed, auxetic and non-auxetic materials display different deformation mechanisms. Revealing these differences and their effects on the mechanics of these materials is of a significant importance. In this study, effects of Poisson ratio on the mechanics of auxetic and non-auxetic nanobeams are revealed. A parametric study is provided on effects of Poisson ratio on the static bending and free vibration behaviors of auxetic nanobeams. The general nonlocal theory is employed to model the nonlocal effects. Unlike Eringen nonlocal theory, the general nonlocal theory uses different attenuation functions for the longitudinal and lateral strains. This theory can reveal the Poisson ratio-nonlocal coupling effects on the mechanics of nanomaterials. The obtained results showed that Poisson ratio is an essential parameter for determining mechanical behaviors of nanobeams. It is revealed that auxetic and non-auxetic nanobeams may reflect softening or hardening behaviors depending on the ratio of the nonlocal fields of the longitudinal and lateral strains of the beam.

A general nonlocal theory and its approximations for slowly varying acoustic waves

In this study, the mechanics of particles, the mechanics of linear elastic continua, and the materials dispersion relations are considered in the framework of a general nonlocal theory which considers different attenuation functions for the distinct material coefficients. For slowly varying acoustic waves, i.e., weak nonlocal fields, this general nonlocal theory is reduced to the strain gradient theory and the couple stress theory. When compared to the general nonlocal theory, the nonlocal characters of the strain gradient and couple stress theories are discovered. Moreover, by fitting the experimental dispersion curves of materials, the nonlocal parameters and the material coefficients and length scales of these theories are reported for materials including diamond, graphite, silicon, silver, gold, copper, and platinum. It is revealed that, when the proper values of the material coefficients are determined, the strain gradient and the couple stress theory can capture the same phenomena as the nonlocal theory. Moreover, it is demonstrated that neither the strain gradient theory nor the couple stress theory can be merged with the nonlocal theory in a unified model. This demolishes the concept based on which the higher-order nonlocal strain gradient theory (Lim et al., 2015) was proposed.

New insights on the applicability of Eringen’s nonlocal theory

New insights on the applicability of Eringen’s nonlocal theory are introduced. Eringen’s nonlocal theory which assumes the same attenuation function for all material moduli is assessed according to its ability to reflect the exact dispersions of different materials. It is revealed that Eringen’s nonlocal theory fails to, simultaneously, fit both the longitudinal and transverse (shear) acoustic dispersion curves of some materials, such as Si, Au, and Pt. In order to extend the limits of applicability of Eringen’s nonlocal theory, a general form of the nonlocal theory is developed. This general nonlocal theory considers different attenuation functions for the distinct material moduli. This makes the proposed nonlocal theory easily fits both the longitudinal and transverse acoustic dispersion curves of different materials. Moreover, unlike Eringen’s conventional form of nonlocal theory, the general nonlocal theory can reflect both hardening and softening behaviors of the material.Another contribution in this study is the determination of the nonlocal parameters for various crystal materials by fitting their dispersions with the proposed nonlocal theory. Thus, the elastic moduli and the nonlocal parameters of metallic crystals (such as Au, Ag, Cu, and Pt), diamond structure crystals (such as diamond and Si), layered structure crystals (such as graphite), Metal Oxides and Alkali Halides with Rock-Salt structures (such as BaO, NiO, LiH and LiD), and Fluorite structure crystals (such as Mg2Sn and Mg2Pb) are reported.

Free energies in a general non-local theory of a material with memory

A general theory of non-local materials, with linear constitutive equations and memory effects, is developed within a thermodynamic framework. Several free energy and dissipation functionals are constructed and explored. These include an expression for the minimum free energy and a functional that is a free energy for important categories of memory kernels and is explicitly a functional of the minimal state. The functionals discussed have a similar general form to the corresponding expressions for simple materials. A number of new results are derived for them, most of which apply equally to both types of material. In particular, detailed formulae are given for these quantities in the case of sinusoidal histories.

A general nonlocal beam theory: Its application to nanobeam bending, buckling and vibration

In the present study, a generalized nonlocal beam theory is proposed to study bending, buckling and free vibration of nanobeams. Nonlocal constitutive equations of Eringen are used in the formulations. After deriving governing equations, different beam theories including those of Euler–Bernoulli, Timoshenko, Reddy, Levinson and Aydogdu [Compos. Struct., 89 (2009) 94] are used as a special case in the present compact formulation without repeating derivation of governing equations each time. Effect of nonlocality and length of beams are investigated in detail for each considered problem. Present solutions can be used for the static and dynamic analyses of single-walled carbon nanotubes.

Optical Spatial Solitons in Soft Matter

We predict that spatial self-trapping of light can occur in soft matter encompassing a wide class of new materials such as colloids, foams, gels, fractal aggregates, etc. We develop a general nonlocal theory that allows one to relate the properties of the trapped state of Maxwell equations to the measurable static structure factor of the specific material. We give numerical evidence for stable trapping in fractal aggregates and suggest also the possibility of soliton spectroscopy of soft matter.

Some relationships within the nonlocal (pair–site) chemical reactivity formalism of density functional theory

General and exact relationships within the nonlocal (pair–site) reactivity context are explicitly derived in the framework of density functional theory. The equations are expressed in terms of linear response functions and associated kernels of the Fukui function and its derivatives with respect to the electron number at constant external potential. It is shown that under a local approximation for the general nonlocal pair–site kernels, the nonlocality character of the chemical response must appear only implicitly through the changes in the chemical potential of the system. In the framework of a more general nonlocal reactivity theory, new relations can be written through the incorporation of more spatial coordinates. The case of a three-site equation between Fukui functions has been explored.

Heat flux near critical density in laser-plasma interaction

A general nonlocal theory of heat transport in laser-plasma interaction experiments is presented. It is shown that the inverse bremsstrahlung absorption and the plasma expansion are responsible for heat fluxes q ib and q h which add to the heat flux due to the temperature gradient itself.