Eringen's Differential Model Research Articles

Differences between stress-driven nonlocal integral model and Eringen differential model in the vibrations analysis of carbon nanotubes conveying magnetic nanoflow

Differences between stress-driven nonlocal integral model and Eringen differential model in the vibrations analysis of carbon nanotubes conveying magnetic nanoflow

An integral nonlocal model for the free vibration analysis of Mindlin nanoplates using the VDQ method

Eringen’s nonlocal theory (ENT) provides an efficient tool to consider the size dependency for the structural analysis of small-scale structures. Recently, it has been indicated that in some cases, inconsistent results may be obtained using the Eringen differential model (EDM) and Eringen integral model (EIM) should be employed. In this regard, the vibration of nanoplates resting on the elastic medium is investigated on the basis of the EIM using variational differential quadrature (VDQ) method. On the basis of the first-order shear deformation theory (FSDT), the EIM and EDM formulations are given. An effective numerical technique is applied in the framework of the energy method to find the natural frequencies. Comprehensive results are reported to compare the EIM and EDM. The results show that using the proposed integral model, the paradox related to cantilever nanoplates is resolved.

Geometrically nonlinear analysis of Timoshenko piezoelectric nanobeams with flexoelectricity effect based on Eringen's differential model

• Extended theory of piezoelectricity is used to investigate flexoelectricity effect on vibrational behavior of nanobeam. • The von-Kármán strain–displacement relationship, electric Gibbs free energy and Hamilton's principle are employed. • Multiple scales method is used to obtain a closed form solution of derived nonlinear governing equations of motion. • The effects of changes in parameters on free vibration and postbuckling behavior of nanobeam are discussed comprehensively. • Theoretical range of flexoelectricity constant is estimated through buckling analysis . This study analyzes the nonlinear free vibration and post-buckling of nanobeams with flexoelectric effect based on Eringen's differential model. The nanobeam is modeled based on Timoshenko beam's theory. The von-Kármán strain–displacement relation together with the electrical Gibbs free energy and Hamilton's principle are employed to derive equations of motion. The nonlinear free vibration frequencies are obtained for pinned–pinned (P–P) and clamped–clamped (C–C) boundary conditions. Multiple scales method is employed to obtain the closed-form solution for the nonlinear governing equations. By employing this methodology, the natural frequencies of nanobeams are obtained and their post-buckling behavior is examined. The influence of nonlocal parameter, amplitude ratio, and input voltage on the top surface and flexoelectricity constant on nonlinear free vibration and post-buckling characteristics of nanobeam is investigated. In this paper, it is concluded that the flexoelectricity has a significant effect on free vibration of the beams in nano-scale and its effect has to be considered in designing nano-electro-mechanical systems (NEMS) such as nano- generators and nano-sensors.

Nonlinear vibration analysis of FG nano-beams resting on elastic foundation in thermal environment using stress-driven nonlocal integral model

This paper comprehensively studies the nonlinear vibration of functionally graded nano-beams resting on elastic foundation and subjected to uniform temperature rise. The small-size effect, playing an essential role in the dynamical behavior of nano-beams, is considered here applying the innovative stress driven nonlocal integral model due to Romano and Barretta. The governing partial differential equations are derived from the Bernoulli–Euler beam theory utilizing the von Karman strain–displacement relations. Using the Galerkin method, the governing equations are reduced to a nonlinear ordinary differential equation. The closed form analytical solution of the nonlinear natural frequency for four different boundary conditions is then established employing the Homotopy Analysis Method. The nonlinear natural frequencies, evaluated according to the stress-driven nonlocal integral model, are compared with those obtained by Eringen differential model. Finally, the effects of different parameters such as length, elastic foundation parameter, thermal loading and nonlocal characteristic parameter are investigated. The emergent results establish that when the nonlocal characteristic parameter increases, the nonlinear natural frequencies obtained by the stress-driven nonlocal integral model reveal a stiffness-hardening effect. On the other hand, Eringen's differential law reveals a stiffness-softening effect excepting the case of cantilever nano-beam. Also, increase in temperature and the elastic foundation parameter leads to increase in the nonlinear frequency ratios in Eringen differential model but decrease in the frequency ratios in the stress-driven nonlocal integral model.

Static and dynamic behaviour of nonlocal elastic bar using integral strain-based and peridynamic models

The static and dynamic behaviour of a nonlocal bar of finite length is studied in this paper. The nonlocal integral models considered in this paper are strain-based and relative displacement-based nonlocal models; the latter one is also labelled as a peridynamic model. For infinite media, and for sufficiently smooth displacement fields, both integral nonlocal models can be equivalent, assuming some kernel correspondence rules. For infinite media (or finite media with extended reflection rules), it is also shown that Eringen's differential model can be reformulated into a consistent strain-based integral nonlocal model with exponential kernel, or into a relative displacement-based integral nonlocal model with a modified exponential kernel. A finite bar in uniform tension is considered as a paradigmatic static case. The strain-based nonlocal behaviour of this bar in tension is analyzed for different kernels available in the literature. It is shown that the kernel has to fulfil some normalization and end compatibility conditions in order to preserve the uniform strain field associated with this homogeneous stress state. Such a kernel can be built by combining a local and a nonlocal strain measure with compatible boundary conditions, or by extending the domain outside its finite size while preserving some kinematic compatibility conditions. The same results are shown for the nonlocal peridynamic bar where a homogeneous strain field is also analytically obtained in the elastic bar for consistent compatible kinematic boundary conditions at the vicinity of the end conditions. The results are extended to the vibration of a fixed–fixed finite bar where the natural frequencies are calculated for both the strain-based and the peridynamic models.