Academic literature on the topic 'Eringen's nonlocal elastica'

The small scale effect on the transient analysis of nanoscale plates is studied. The elastic theory of the nano-scale plate is reformulated using Eringen’s nonlocal differential constitutive relations and higher-order shear deformation theory (HSDT). The equations of motion of the nonlocal theories are derived for the nano-scale plates. The Eringen’s nonlocal elasticity of Eringen has ability to capture the small scale effects and the higher-order shear deformation theory has ability to capture the quadratic variation of shear strain and consequently shear stress through the plate thickness. The solutions of transient dynamic analysis of nano-scale plate are presented using these theories to illustrate the effect of nonlocal theory on dynamic response of the nano-scale plates. On the basis of those numerical results, the relations between nonlocal and local theory are investigated and discussed, as are the nonlocal parameter, aspect ratio, side-to-thickness ratio, nano-scale plate size, and time step effects on the dynamic response. In order to validate the present solutions, the reference solutions are employed and examined. The results of nano-scale plates using the nonlocal theory can be used as a benchmark test for the transient analysis.

This article investigates vibration behavior of magneto-electro-elastic functionally graded nanobeams embedded in two-parameter elastic foundation using a third-order parabolic shear deformation beam theory. Material properties of magneto-electro-elastic functionally graded nanobeam are supposed to be variable throughout the thickness based on power-law model. Based on Eringen’s nonlocal elasticity theory which captures the small size effects and using Hamilton’s principle, the nonlocal governing equations of motions are derived and then solved analytically. Then, the influences of elastic foundation, magnetic potential, external electric voltage, nonlocal parameter, power-law index, and slenderness ratio on the frequencies of the embedded magneto-electro-elastic functionally graded nanobeams are studied.

Several of the Eringen’s nonlocal stress models, including two-phase and purely nonlocal integral models, along with the simplified differential model, are studied in the case of free longitudinal vibrations of a nanorod, for various types of boundary conditions. Assuming the exponential attenuation kernel in the nonlocal integral models, the integro-differential equation corresponding to the two-phase nonlocal model is reduced to a fourth-order differential equation with additional boundary conditions, taking into account nonlocal effects in the neighbourhood of the rod ends. Exact analytical and asymptotic solutions of boundary-value problems are constructed. Formulas for natural frequencies and associated modes found in the framework of the purely nonlocal model and its ‘equivalent’ differential analogue are also compared. A detailed analysis of solutions suggests that the purely nonlocal and differential models lead to ill-posed problems.

Evaluation of size effects in functionally graded elastic nanobeams is carried out by making recourse to the nonlocal continuum mechanics. The Bernoulli-Euler kinematic assumption and the Eringen nonlocal constitutive law are assumed in the formulation of the elastic equilibrium problem. An innovative methodology, characterized by a lowering in the order of governing differential equation, is adopted in the present manuscript in order to solve the boundary value problem of a nanobeam under flexure. Unlike standard treatments, a second-order differential equation of nonlocal equilibrium elastic is integrated in terms of transverse displacements and equilibrated bending moments. Benchmark examples are developed, thus providing the nonlocality effect in nanocantilever and clampled-simply supported nanobeams for selected values of the Eringen scale parameter.

The free vibrations of elastic circular arches made of a nano-material are considered. A method of determination of eigenfrequencies of nanoarches weakened with stable cracks is developed making use of the concept of the massless spring and Eringen's nonlocal theory of elasticity. The aim of the paper is to evaluate the sensitivity of eigenfrequencies on the geometrical and physical parameters of the nanoarch.

AbstractIn this study, nonlinear wave modulation in nanorods is examined on the basis of nonlocal elasticity theory. Eringen's nonlocal elasticity theory is employed to derive nonlinear equations for the motion of nanorods. The analysis of the modulation of axial waves in nonlocal elastic media is performed, and the reductive perturbation method is used for the solution of the nonlinear equations. The propagation of weakly nonlinear and strongly dispersive waves is investigated, and the nonlinear Schrödinger (NLS) equation is acquired as an evolution equation. For the purpose of a numerical investigation of the nonlocal impacts on the NLS equation, it has been investigated whether envelope solitary wave solutions exist by utilizing the physical and geometric features of the carbon nanotubes. Amplitude dependent wave frequencies, phase and group velocities have been obtained and they have compared for the linear local, the linear nonlocal, the nonlinear local and the nonlinear nonlocal cases.

After adopting the constitutive equations of the nonlocal elastic media in the form of Eringen, and making use of the Laplace transformation, the vibration governing equation of nonlocal elastic beam in the Kelvin media are established. Unlike classical elastic models, the stress of a point in a nonlocal model is obtained as a weighted average of the field over the spatial domain, determined by a kernel function based on distance measures. The motion equation of nonlocal elastic beam is an integral differential equation, rather than the differential equation obtained with a classical local model. Solutions for natural frequencies and modes are obtained. Numerical examples demonstrate the efficiency of the proposed method for the beam with simple boundary conditions.

Investigated herein is the postbuckling behavior of an initially imperfect nonlocal elastic column, which is simply supported at one end and subjected to an axial force at the other movable end. The governing nonlinear differential equation of the axially loaded nonlocal elastic column experiencing large deflection is first established within the framework of Eringen's nonlocal elasticity theory in order to embrace the size effect. Its semianalytical solutions by the virtue of homotopy perturbation method, as well as the successive approximation algorithm, are determined in an explicit form, through which the postbuckling equilibrium loads in terms of the end rotation angle and the deformed configuration of the column at this end rotation are predicted. By comparing the degenerated results with the exact solutions available in the literature, the validity and accuracy of the proposed methods are numerically substantiated. The size effect, as well as the initial imperfection, on the buckled configuration and the postbuckling equilibrium path is also thoroughly discussed through parametric studies.

In this research, free vibration behavior of thick functionally graded nanoplates is carried out using the Chebyshev spectral collocation method. It is assumed that the plates are resting on variable elastic foundations. Eringen’s nonlocal elasticity theory is used to capture the size effect, and Mindlin’s first-order shear deformation plate theory is employed to model the thick nanoplates. Hamilton’s principle along with the differential form of Eringen’s constitutive relations are utilized to obtain the governing partial differential equations of motion for the functionally graded nanoplates under consideration. A numerical solution is presented by applying the spectral collocation method and the natural frequencies are obtained. A parametric study is conducted to study the effects of several factors on the natural frequencies of the functionally graded nanoplates. It is found that the parameters of the variable elastic foundation (Winkler and shear moduli), thickness to length ratio, length to width ratio (aspect ratio), the nonlocal scale coefficient, the gradient index, the foundation type, and the boundary conditions have a remarkable influence on the free vibration characteristics of the functionally graded nanoplates.

Based on a nonlocal elasticity theory, a model for sigmoid functionally graded material (S-FGM) nanoscale plate with first-order shear deformation is studied. The material properties of S-FGM nanoscale plate are assumed to vary according to sigmoid function (two power law distribution) of the volume fraction of the constituents. Elastic theory of the sigmoid FGM (S-FGM) nanoscale plate is reformulated using the nonlocal differential constitutive relations of Eringen and first-order shear deformation theory. The equations of motion of the nonlocal theories are derived using Hamilton’s principle. The nonlocal elasticity of Eringen has the ability to capture the small scale effect. The solutions of S-FGM nanoscale plate are presented to illustrate the effect of nonlocal theory on bending and vibration response of the S-FGM nanoscale plates. The effects of nonlocal parameters, power law index, aspect ratio, elastic modulus ratio, side-to-thickness ratio, and loading type on bending and vibration response are investigated. Results of the present theory show a good agreement with the reference solutions. These results can be used for evaluating the reliability of size-dependent S-FGM nanoscale plate models developed in the future.

En mathématiques, le calcul des variations est un ensemble de méthodes permettant la détermination de solutions à des problèmes d'optimisation des quantités traduites en termes de fonctionnelle. De nombreuses applications existent, notamment dans la recherche de courbes ou de surfaces minimales. Les systèmes dynamiques considérés sont de natures diverses (équations différentielles, intégrales ou stochastiques) et modélisent des problèmes d'origines multiples : aérospatiale, automobile, biologie, économie, médecine, etc. Le théorème de Noether présente un fort intérêt puisqu'il propose une loi de conservation explicite (traduisant souvent une quantité physique comme l'énergie totale ou le moment angulaire en mécanique classique) qui permet de réduire ou d'intégrer l'équation différentielle associée par quadrature. L'objectif de ma thèse contient de nombreux thèmes, dans le premier but nous allons : *) donner le théorème de Noether discret dans le cadre ”time scale” (Le formalisme lagrangien et hamiltonien). Le passage de la nature discrète à la nature continue de la structure la morphologie est d'un intérêt primordial en physique pour comprendre comment la microstructure peut influencer les propriétés macroscopiques du matériau à plus grande échelle. Ce passage peut être modélisé par un système discret appelé 'Hencky's chain' et l'équation du mouvement est donnée par des équations aux différences non linéaires et cette équation ne possède pas de Lagrangien. Le deuxième but nous allons : *) donner les structures lagrangienne, hamiltonienne via le facteur intégrant et trouver la solution analytique de l'équation non locale au sens d'Eringen (nonlocalité différentielle d'Eringen, 1983). Le troisième but nous allons : *) étudier l'existence des formulations variationnelles via le principe de Brezis Ekeland-Nayroles (Gery de Saxce) - application sur la formulation 4D développée par E. Rouhaud pour l'étude des déformations des matériaux *) développer des schémas numériques qui respectent certaines particularités. En particulier, un schéma permettent de mettre en œuvre la théorie 4D développée par E. Rouhaud. *) applications numériques et théoriques sur le problème des déformations des matériaux
The aim of this thesis is to deal with the connection between continuous and discrete versions of a given object. This connection can be studied in two different directions: one going from a continuous setting to a discrete analogue, and in a symmetric way, from a discrete setting to a continuous one. The first procedure is typically used in numerical analysis in order to construct numerical integrators and the second one is typical of continuous modeling for the study of micro-structured materials.In this manuscript, we focus our attention on three distinct problems. In the first part, we propose a general framework precising different ways to derive a discrete version of a differential equation called discrete embedding formalism.More precisely, we exhibit three main discrete associate: the differential, integral or variational structure in both classical and high-order approximations.The second part focuses on the preservation of symmetries for discrete versions of Lagrangian and Hamiltonian systems, i.e., the discrete analogue of Noether's theorem.Finally, the third part applies these results in mechanics, i.e., the problem studied by N. Challamel, Kocsis and Wang called Eringen's nonlocal elastica equation which can beobtained by the continualization method. Precisely, we construct a discrete version of Eringen's nonlocal elastica then we study the difference with Challamel's proposal

In this study, the primary response of an Euler-Bernoulli beam resting on nonlinear elastic foundation is investigated. The beam is subjected to thermal and magnetic axial loads. The nonlocal Eringen’s elasticity theory is used to derive the mathematical model to account for the scale effect of the beam. A simply supported beam is considered in the analysis, and the multi-mode approach is used to obtain the reduced nonlinear temporal equations of motion that contain quadratic and cubic nonlinear terms. The method of multiple-scales is applied to obtain approximate analytical solutions for the nonlinear natural frequencies in addition to the primary resonance response curves. Moreover, the effective nonlinearity is obtained as a function of the natural frequencies and the coefficients of the elastic foundation. The results reveal that the scale parameter has a significant effect on the frequencies and amplitudes of the beam. The obtained results are presented over a selected range of physical parameters such as the scale effect parameter, foundation parameters, thermal and magnetic loads, and the excitation level. Time responses, phase planes and Poincaré maps are generated for the beam under consideration.