A new nonlocal bending model for Euler–Bernoulli nanobeams

This paper is concerned with the bending problem of nanobeams starting from a nonlocal thermodynamic approach. A new coupled nonlocal model, depending on two nonlocal parameters, is obtained by using a suitable definition of the free energy. Unlike previous approaches which directly substitute the expression of the nonlocal stress into the classical equilibrium equations, the proposed approach provides a methodology to recover nonlocal models starting from the free energy function. The coupled model can then be specialized to obtain a nanobeam formulation based on the Eringen nonlocal elasticity theory and on the gradient elastic model. The variational formulations are consistently provided and the differential equations with the related boundary conditions are thus derived. Nanocantilevers are solved in a closed-form and numerical results are presented to investigate the influence of the nonlocal parameters.     

A nonlocal phase-field model of Ginzburg–Landau–Korteweg fluids

A thermodynamic model of Korteweg fluids undergoing phase transition and/or phase separation is developed within the framework of weakly nonlocal thermodynamics. Compatibility with second law of thermodynamics is investigated by applying a generalized Liu procedure recently introduced in the literature. Possible forms of the free energy and of the stress tensor, which generalize some earlier ones proposed by several authors in the last decades, are carried out. Owing to the new procedure applied for exploiting the entropy principle, the thermodynamic potentials are allowed to depend on the whole set of variables spanning the state space, including the gradients of the unknown fields, without postulating neither the presence of an energy or entropy extra-flux, nor an additional balance law for microforce.     

An asymmetric theory of nonlocal elasticity—Part 1. Quasicontinuum theory

In this article, an asymmetric theory of nonlocal elasticity is developed on the basis of three dimensional atomic lattice model, the Galileo invariance for constitutive equations and by use of Fourier transformation of generalized function and energy method. It is shown that nonlocal characteristic functions (or constitutive parameters of internal elastic energy) can be explicitly expressed in terms of interacting forces connecting atoms, and the general model of nonlocal theory with rotation effects is asymmetric. Both symmetric stress and anti-symmetric stress is a nonlocal function of strain and local rotation for anisotropic materials. For isotropic materials, symmetric stress is only a nonlocal function of strain, while antisymmetric stress is only a nonlocal function of local rotation.     

Wavefront solutions in diffusive Nicholson’s blowflies equation with nonlocal delay

This paper is concerned with the diffusive Nicholson’s blowflies equation with nonlocal delay incorporated as an integral convolution over the entire past time up to now and the whole one-dimensional spatial domain R.Assume that the delay kernel is a strong generic kernel.By the linear chain techniques and the geometric singular perturbation theory,the existence of travelling front solutions is shown for small delay.     

Wavefront solutions in diffusive Nicholson＇s blowflies equation with nonlocal delay

This paper is concerned with the diffusive Nicholson＇s blowflies equation with nonlocal delay incorporated as an integral convolution over the entire past time up to now and the whole one-dimensional spatial domain R. Assume that the delay kernel is a strong generic kernel. By the linear chain techniques and the geometric singular perturbation theory, the existence of travelling front solutions is shown for small delay.     

An asymmetric theory of nonlocal elasticity—Part 2. Continuum field

In this paper, an asymmetric theory of nonlocal elasticity with nonlocal body couple is developed on the basis of the axiom system in nonlocal continuum field theory. The Galileo invariance is used for determining the explicit form of the constitutive equations. It is shown that both continuum field theory and quasicontinuum theory give the same constitutive equations and field equations for the general theory of nonlocal elasticity. Finally, the relations among nonlocal theory, couple stress theory, and higher gradient theory are investigated.     

Theoretical analysis on elastic buckling of nanobeams based on stress-driven nonlocal integral model

Several studies indicate that Eringen's nonlocal model may lead to some inconsistencies for both Euler-Bernoulli and Timoshenko beams, such as cantilever beams subjected to an end point force and fixed-fixed beams subjected a uniform distributed load. In this paper, the elastic buckling behavior of nanobeams, including both EulerBernoulli and Timoshenko beams, is investigated on the basis of a stress-driven nonlocal integral model. The constitutive equations are the Fredholm-type integral equations of the first kind, which can be transformed to the Volterra integral equations of the first kind. With the application of the Laplace transformation, the general solutions of the deflections and bending moments for the Euler-Bernoulli and Timoshenko beams as well as the rotation and shear force for the Timoshenko beams are obtained explicitly with several unknown constants. Considering the boundary conditions and extra constitutive constraints, the characteristic equations are obtained explicitly for the Euler-Bernoulli and Timoshenko beams under different boundary conditions, from which one can determine the critical buckling loads of nanobeams. The effects of the nonlocal parameters and buckling order on the buckling loads of nanobeams are studied numerically, and a consistent toughening effect is obtained.     

Families of nonsingular soliton solutions of a nonlocal Schrödinger–Boussinesq equation

Nonlocal nonlinear evolution equations with self-induced parity–time symmetric potential have received intensive attention, due to their good applications in nonlinear optics. A nonlocal Schrödinger–Boussinesq equation is proposed in this paper. By using the Hirota bilinear method and the Kadomtsev–Petviashvili hierarchy reduction method, explicit soliton solution with the nonzero boundary condition is succinctly constructed in terms of determinant. Typical dynamics and asymptotic behaviours of three types of two-soliton solutions are discussed in detail.     

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.     

On the minimal speed of traveling waves for a nonlocal delayed reaction–diffusion equation

In this note, we give constructive upper and lower bounds for the minimal speed of propagation of traveling waves for a nonlocal delayed reaction–diffusion equation.     

Wave propagation responses of porous bi-directional functionally graded magneto-electro-elastic nanoshells via nonlocal strain gradient theory

This study examines the wave propagation characteristics for a bi-directional functional grading of barium titanate(BaTiO₃) and cobalt ferrite(CoFe₂O₄) porous nanoshells,the porosity distribution of which is simulated by the honeycomb-shaped symmetrical and asymmetrical distribution functions.The nonlocal strain gradient theory(NSGT) and first-order shear deformation theory are used to determine the size effect and shear deformation,respectively.Nonlocal governin...     

Towards a unified nonlocal,peridynamics framework for the coarse-graining of molecular dynamics data with fractures

Molecular dynamics(MD) has served as a powerful tool for designing materials with reduced reliance on laboratory testing. However, the use of MD directly to treat the deformation and failure of materials at the mesoscale is still largely beyond reach. In this work, we propose a learning framework to extract a peridynamics model as a mesoscale continuum surrogate from MD simulated material fracture data sets. Firstly,we develop a novel coarse-graining method, to automatically handle the material ...     

Eringen’s length scale coefficient for buckling of nonlocal rectangular plates from microstructured beam-grid model

Based on a microstructured beam-grid model, closed form solutions for Eringen’s length scale coefficient are derived for the buckling of nonlocal rectangular plates under compressive uniaxial load. The microstructured beam-grid model comprises rigid beam elements connected by rotational springs and torsional springs. Stiffness of the rotational springs and torsional springs are expressed in terms of the plate flexural and torsional rigidities, respectively. Exact buckling solutions are obtained for the microstructured beam-grid model. By interpreting the rigid element length as the internal characteristic length, the Eringen’s length scale coefficient $e_0$ may be determined by comparing the buckling load from the microstructured beam-grid model and the nonlocal rectangular plate model. The small length scale coefficient is shown to be dependent on the buckling mode, the aspect ratio of plate and boundary conditions.     

Semi-analytic solution of Eringen’s two-phase local/nonlocal model for Euler-Bernoulli beam with axial force

Eringen’s two-phase local/nonlocal model is applied to an Euler-Bernoulli nanobeam considering the bending-induced axial force, where the contribution of the axial force to bending moment is calculated on the deformed state. Basic equations for the corresponding one-dimensional beam problem are obtained by degenerating from the three-dimensional nonlocal elastic equations. Semi-analytic solutions are then presented for a clamped-clamped beam subject to a concentrated force and a uniformly distributed load, respectively. Except for the traditional essential boundary conditions and those required to be satisfied by transferring an integral equation to its equivalent differential form, additional boundary conditions are needed and should be chosen with great caution, since numerical results reveal that non-unique solutions might exist for a nonlinear problem if inappropriate boundary conditions are used. The validity of the solutions is examined by plotting both sides of the original integro-differential governing equation of deflection and studying the error between both sides. Besides, an increase in the internal characteristic length would cause an increase in the deflection and axial force of the beam.     

Dynamic stability analysis of porous functionally graded beams under hygro-thermal loading using nonlocal strain gradient integral model

We present a study on the dynamic stability of porous functionally graded(PFG) beams under hygro-thermal loading. The variations of the properties of the beams across the beam thicknesses are described by the power-law model. Unlike most studies on this topic, we consider both the bending deformation of the beams and the hygro-thermal load as size-dependent, simultaneously, by adopting the equivalent differential forms of the well-posed nonlocal strain gradient integral theory(NSGIT) which are s...     

On nonconservativeness of Eringen’s nonlocal elasticity in beam mechanics: correction from a discrete-based approach

In this paper, the self-adjointness of Eringen’s nonlocal elasticity is investigated based on simple one-dimensional beam models. It is shown that Eringen’s model may be nonself-adjoint and that it can result in an unexpected stiffening effect for a cantilever’s fundamental vibration frequency with respect to increasing Eringen’s small length scale coefficient. This is clearly inconsistent with the softening results of all other boundary conditions as well as the higher vibration modes of a cantilever beam. By using a (discrete) microstructured beam model, we demonstrate that the vibration frequencies obtained decrease with respect to an increase in the small length scale parameter. Furthermore, the microstructured beam model is consistently approximated by Eringen’s nonlocal model for an equivalent set of beam equations in conjunction with variationally based boundary conditions (conservative elastic model). An equivalence principle is shown between the Hamiltonian of the microstructured system and the one of the nonlocal continuous beam system. We then offer a remedy for the special case of the cantilever beam by tweaking the boundary condition for the bending moment of a free end based on the microstructured model.     

A general nonlocal nonlinear Schrödinger equation with shifted parity,charge-conjugate and delayed time reversal

A general nonlocal nonlinear Schrödinger equation with shifted parity, charge-conjugate and delayed time reversal is derived from the nonlinear inviscid dissipative and equivalent barotropic vorticity equation in a \(\beta \)-plane. The modulational instability (MI) of the obtained system is studied, which reveals a number of possibilities for the MI regions due to the generalized dispersion relation that relates the frequency and wavenumber of the modulating perturbations. Exact periodic solutions in terms of Jacobi elliptic functions are obtained, which, in the limit of the modulus approaches unity, reduce to soliton, kink solutions and their linear superpositions. Representative profiles of different nonlinear wave excitations are displayed graphically. These solutions can be used to model different blocking events in climate disasters. As an illustration, a special approximate solution is given to describe a kind of two correlated dipole blocking events.     

Higher-dimensional soliton structures of a variable-coefficient Gross–Pitaevskii equation with the partially nonlocal nonlinearity under a harmonic potential

Nonlinear Dynamics - A reduction correlation between the $$(3+1)$$ -dimensional variable-coefficient Gross–Pitaevskii equation with the partially nonlocal nonlinearity under a harmonic...