Completing Eringen’s nonlocal elasticity theory and its connection with surface elasticity

Surface energy effects on thermoelastic vibration of nanomechanical systems under Moore–Gibson–Thompson thermoelasticity and Eringen’s nonlocal elasticity theories

This paper evaluates the performance of local and nonlocal integral elasticity theories at the nanoscale by comparing them with molecular dynamics (MD) simulations. The study investigates the validity of local elasticity theory at the nanoscale and assesses the effectiveness of nonlocal formulations in capturing material behavior at this scale. Within the nonlocal framework, two-phase, modified, and compensation two-phase (CTP) kernels are employed, and their constitutive equations are implemented using Abaqus user-defined subroutines within the finite element method. To effectively compare local and nonlocal theories, a crack opening problem under limited loading, which preserves elastic material behavior, is analyzed. This setup, characterized by pronounced strain gradients and free-boundary conditions near the crack tip, highlights the differences between nonlocal kernels, particularly near boundaries where their behavior diverges despite similar responses in the interior. BCC iron is chosen for its widespread applications, and crack evolution results are compared against MD simulations. In a cracked single crystal (i.e. homogeneous domain), local elasticity aligns well with the MD results. However, when the model is adapted to a cracked bicrystal with a Σ5(310) symmetric grain boundary near the crack tip, local elasticity fails to capture the correct behavior. In contrast, nonlocal elasticity with all three kernels and proper values of nonlocal and phase parameters yields better agreement with the MD results, and the effect of each parameter on their accuracy is analyzed in detail. Among the kernels, the CTP kernel consistently produces the smoothest crack edge and offers more uniform, physically consistent results, regardless of parameter variation. Conversely, the modified kernel leads to less favorable crack morphology, sometimes causing the crack edges to tilt outward near the tip. Quantitative comparison with MD results using the nodal deviations indicates that nonlocal integral models generally improve accuracy over the local formulation. The CTP kernel achieves up to 6.28% reduction in the total nodal deviation for the sharp crack and up to 10.55% for the blunt crack, with the improvement strongly dependent on the selected nonlocal and phase parameters.

Finite Element Model and Size Dependent Stability Analysis of Boron Nitride and Silicon Carbide Nanowires/Nanotubes

Torsional instability of carbon nano-peapods based on the nonlocal elastic shell theory

Buckling analysis of a single-walled carbon nanotube embedded in an elastic medium based on nonlocal elasticity and Timoshenko beam theory and using DQM

In this paper, buckling analysis of biaxially compressed graphene sheets with non-local elasticity theory is reported. The equations of motion for graphene sheet are derived using non-local local elasticity theory. Levy’s approach has been used to solve the governing equations for various boundary conditions of the graphene sheet. Present results from Levy’s solution agree with the results for all edges simply supported available in the literature. Further, the effect of the (i) non-local parameter, (ii) size of the graphene sheet and (iii) various boundary conditions on the critical buckling loads of the graphene sheets are investigated. It is observed that non-local parameter and boundary conditions significantly influence the critical buckling loads of the small size graphene sheets.

Thermoelastic damping has emerged as a critical issue in the modeling and design of micro and nanomechanical systems. Therefore, an extensive research interest is being devoted towards the reduction in thermoelastic damping for micro and nanomechanical systems. The present study intends to examine thermoelastic damping in nanobeam resonators using the modified couple stress theory and Eringen’s nonlocal elasticity theory, thus so-called modified nonlocal couple stress theory within the context of the recently proposed Moore–Gibson–Thompson thermoelasticity theory. In order to observe size effects, the size-dependent coupled thermoelastic equations are derived by combining modified couple stress theory and nonlocal elasticity theory in the frame of Moore–Gibson–Thompson thermoelasticity. The coupled governing equations for thermoelastic damping of nanobeam resonators are solved analytically in terms of the inverse quality factor. The size-dependent thermoelastic damping of nanobeams is presented graphically with the help of numerical results predicted by modified nonlocal couple stress theory. The results obtained under the combined effects of modified couple stress theory and nonlocal theory in the present context are further compared to those of the classical, modified couple stress, and nonlocal elasticity theories as special cases of the modified nonlocal couple stress theory. It is observed that thermoelastic damping weakens at the submicron scale under Eringen’s nonlocal elasticity theory, whereas it becomes greater at the submicron scale under modified couple stress theory. However, at the submicron scale, the modeling of nanobeam resonators using modified nonlocal couple stress theory reveals a smaller amount of thermoelastic damping than modified couple stress, nonlocal, and classical theories. In addition, as compared to the Green–Naghdi thermoelasticity theory of type III (GN-III), the Moore–Gibson–Thompson model predicts a higher thermoelastic damping value, while agreeing with the results predicted by the Lord–Shulman (LS) model under modified nonlocal couple stress theory. Some important points highlighted here are believed to be useful in the design of mechanical resonators at micron and submicron scales.

Nonlinear bending analysis of bilayer orthotropic graphene sheets resting on Winkler–Pasternak elastic foundation based on non-local continuum mechanics

Buckling of single layer graphene sheet based on nonlocal elasticity and higher order shear deformation theory

A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation

In this paper, a detailed theoretical study on the dispersion of waves in carbon nanotubes (CNTs) is presented. For this purpose, CNTs are considered as nonlocal elastic thin cylindrical shells. The Eringen’s nonlocal elasticity theory is used for modeling the microstructure of CNT such that the proximity of the mathematical model to the actual atomic structure of CNT is retained. The results are compared with the results that are obtained based on the second-order strain-gradient elasticity (SG) theory. It has been shown that the SG theory is the first approximation of nonlocal continuum elasticity (NC) theory, which is used in the present paper. Also, it has been shown that the bending rigidity has important effect in the dispersion of waves in CNTs.

Nonlocal elasticity theory is a growing technique for the mechanical analyses of microelectromechanical (MEMS) and nanoelectromechanical (NEMS) based structures. The nonlocal parameter accounts for the small size effects when dealing with nanosize structures such as single-walled carbon nanotubes (SWCNTs). In this article, nonlocal elasticity and Timoshenko beam theory are implemented to study the vibration response of SWCNT embedded in an elastic medium. Influence of the surrounding elastic medium on the fundamental frequencies of the SWCNT is investigated. Both Winkler-type and Pasternak-type foundation models are employed to simulate the interaction of the SWCNT with the surrounding elastic medium. A differential quadrature approach is being utilized and numerical solutions for the natural frequencies are obtained. Influences of nonlocal effects, Winkler modulus parameter, Pasternak shear modulus parameter, and aspect ratio on the frequency of SWCNT are analyzed and discussed. The present study illustrates that the frequencies of embedded SWCNT are significantly dependent on the nonlocal parameter and on the stiffness of the surrounding elastic medium.

Polysilicon nano-beam model based on modified couple stress and Eringen’s nonlocal elasticity theories

Forced vibrations of the single wall carbon nanotubes (SWCNTs) using the non-local elasticity theory and Timoshenko beam theory is studied. Governing equations for the vibration response of the carbon nanotubes are derived employing Eringen’s nonlocal elasticity theory. Analysis is carried out for both EulerBernoulli beam theory and Timoshenko beam theory. Present forced vibration results for Euler-Bernoulli are in good agreement with those reported in literature. Effects of (i) nonlocal parameter (ii) length of the carbon nanotube and (iii) modes of vibration on the vibration response of the carbon nanotubes are investigated. The effect of nonlocal parameter on the vibration response are significant for small size carbon nanotube and higher modes of vibration.

The vibration behavior of size-dependent nano-crystalline nano-beams is investigated based on nonlocal, couple stress and surface elasticity theories. A nanocrystalline nano-beam is composed of three phases which are nano-grains, nano-voids, and interface. Nano-voids or porosities inside the material have a stiffness-softening impact on the nano-beam. A Eringen’s nonlocal elasticity theory is applied in the analysis of nano-crystalline nano-beams for the first time. Residual surface stresses which are usually neglected in modeling nano-crystalline nano-beams are incorporated into nonlocal elasticity to better understand the physics of the problem. Also, a modified couple stress theory is used to capture rigid rotations of grains. Applying a differential transform method (DTM) satisfying various boundary conditions, the governing equations obtained from the Hamilton’s principle are solved. Reliability of the proposed approach is verified by comparing the obtained results with those provided in the literature. The effects of the nonlocal parameter, surface effect, couple stress, grain size, porosities, and interface thickness on the vibration characteristics of nano-crystalline nano-beams are explored.