Nonlocal Elasticity Research Articles

A novel dynamic stiffness matrix for the nonlocal vibration characteristics of porous functionally graded nanoplates on elastic foundation with small-scale effects

The present paper deals with the development of a dynamic stiffness matrix to evaluate the free vibration response of functionally graded nanoplate (FG-nP) resting on the Winkler-Pasternak elastic foundation. The complete mathematical modeling of the dynamic stiffness matrix for nanostructures is given for the first time. The equation of motion for rectangular FG-nP plates supported on an elastic foundation is derived using Hamilton’s principle in conjunction with nonlocal elasticity theory. The non-local theory is incorporated to account for the size effect in the small-scale plate. The effective material property of the porous FG-nP has been calculated using three recently developed models of porosity. The developed dynamic stiffness matrix is solved using the Wittrick-Williams algorithm to extract the natural frequencies of the FG-nP. The variation of natural frequencies with the change of numerical values, such as nonlocal parameter, aspect ratio, elastic foundation parameters, and porosity volume fraction is analyzed. The validity and accuracy of the results are confirmed through comparison with the available literature. The use of non-local theory in dynamic stiffness analysis is shown to be effective in predicting the natural frequency of the FG-nP on a Winkler-Pasternak elastic foundation, providing new insights into the dynamic behavior of small-scale structures.

High-frequency temperature-dependent vibration of nonlocal functionally graded sandwich nanoplates resting on elastic foundations

This article studies the role of temperature and nonlocality on the low- and high-frequency behaviors of functionally graded sandwich nanoplates resting on Winkler–Pasternak foundations. The small-scale effects are incorporated using nonlocal elasticity theory. The rise in temperature is characterized by the uniform, linear, and nonlinear distributions. The governing equations are solved using Navier’s solution, ensuring convergence of frequencies at any mode sequence number. A comprehensive parametric study is presented to elucidate the effects of various parameters on the low- and high-frequency behaviors of functionally graded sandwich nanoplates. The findings provide fundamental insights for the design, testing, and manufacture of nanostructures.

Nonlinear Wave Interaction of Nanorods Embedded in a Viscoelastic Medium

Abstract Purpose Nonlinear interactions between two acoustic waves in nanorods traveling at various wave numbers, group velocities, and frequencies are examined in this study. Methods The nonlinear equation of the nanorod in a viscoelastic medium is obtained using the theory of nonlocal elasticity. Furthermore, the multiple-scale expansion method is applied to study strongly dispersive, weakly nonlinear waves in a nonlocal viscoelastic medium. Using this expansion technique, we can derive the coupled nonlinear Schrödinger equations as the governing equations, which we solve as differential equations of some parameters by expanding the field quantities into an asymptotic series of the smallness parameter. Results We give the nonlinear plane wave solutions to these equations in several special cases. The plane wave solutions show how the wave amplitude affects the frequencies of nonlinear plane waves. Additionally, we show numerically how the real and imaginary parts of the group velocities and natural frequency of the system for a carbon nanotube in a viscoelastic medium are affected by the nonlocal, damping, and stiffness parameters.

Size-dependent effect on the interaction of surface waves in micropolar thermoelastic medium with dual pore connectivity

This research tackles a critical knowledge gap in Rayleigh surface wave propagation. It offers a comprehensive analysis that surpasses previous limitations. A size-dependent micropolar medium with unique void distributions and thermal effects is considered in this work. The constitutive relations and equations of motion for a nonlocal micropolar thermoelastic medium with double voids (MTMWDV) have been established by using Eringen’s nonlocal elasticity theory. Employing the three-phase-lag thermoelasticity theory (TPLTE), the study utilizes a wave-mode method to derive analytical solutions for Rayleigh waves in a nonlocal MTMWDV. To gain a comprehensive understanding of wave behavior, we solve the characteristic equation and analyze its roots, applying a filter based on the surface wave decay condition. A medium with stress-free and isothermal boundaries is explored through computational simulations to determine the attenuation coefficient and phase velocity. Furthermore, particle motion analysis is conducted to complement the analytical and computational approaches. Moreover, the influence of the nonlocal parameter and various thermoelastic models on these wave phenomena is investigated. The validity of the current mathematical model is confirmed through the derivation of particular scenarios.

Non local analytical and numerical modelling of re-entrant auxetic honeycomb

Auxetic materials, characterized by their negative Poisson’s ratio, have been extensively studied for applications in energy absorption and mechanical reinforcement. Re-entrant honeycomb structures, a subtype of auxetic materials, have demonstrated superior mechanical characteristics. However, understanding the mechanical behavior of these structures at the nanoscale remained a significant challenge. To address this gap, the authors explored the influence of size on the deflection behavior of re-entrant auxetic honeycomb structures through non-local continuum mechanics. Their analytical model, incorporating the Euler-Bernoulli beam model and considering four non-dimensional geometrical parameters, was validated through numerical simulations and a comprehensive review of existing literature. The study aimed to provide valuable insights into the design and engineering of re-entrant auxetic honeycomb structures across diverse applications, contributing to the advancement of non-local elasticity theory and deepening the understanding of the mechanical behavior of auxetic structures at the nanoscale. The research laid a foundation for further exploration and optimization of re-entrant honeycomb structures, facilitating their effective utilization in fields such as MEMS/NEMS by leveraging the dimensionless parameters identified in the study.

PINN-based forward and inverse bending analysis of nanobeams on a three-parameter nonlinear elastic foundation including hardening and softening effect using nonlocal elasticity theory

PINN-based forward and inverse bending analysis of nanobeams on a three-parameter nonlinear elastic foundation including hardening and softening effect using nonlocal elasticity theory

Magnetic field effects on the thermomechanical vibration behavior of functionally graded biocompatible material sandwich nanobeams

This study focuses on modeling and analyzing the thermomechanical vibration behavior of sandwich nanobeams made from biocompatible zirconia and Ti-6Al-4V materials. The sandwich nanobeam is composed of the materials mentioned earlier, featuring a porous structure in the core layer. The sandwich nanobeam experiences significant thermal load and is subjected to a horizontal external electromagnetic field. The equations of motion for the sandwich nanobeam were derived through the application of the sinusoidal high-order shear stress theorem and the constitutive equation based on nonlocal strain gradient elasticity. The equations were converted into equations of motion using Hamilton’s principle, incorporating the thermal load and the Lorentz force generated in the electromagnetic field, and solved using the Navier method. Analyses were conducted on the thermomechanical vibration behavior of the sandwich nanoplate, taking into consideration the properties of the sandwich layers, the impact of temperature, and the intensity of the applied horizontal magnetic field. The impact of the porous core layer in the sandwich nanoplate on the porosity volume ratio and porosity distribution function in thermomechanical behavior has been established. Research has demonstrated that the application of an external magnetic field can effectively mitigate the adverse impacts of thermal loads caused by high temperatures.

WITHDRAWN: Accurate and efficient analytical simulation of free vibration for embedded nonlocal CNTRC beams with general boundary conditions

This study aims to evaluate the free vibrational response of restrained nanobeam enriched by nanocomposites based on an exact Fourier series approach. In order to capture the small size effects on the dynamical response, Eringen’s differential form of nonlocal elasticity is used which employs the one size (nonlocal) parameter. Within the framework of Rayleigh and Bernoulli-Euler beam theories to include the effect of nonlocality and by employing Fourier sine series together with Stokes’ transformation, a system of linear equations is obtained then solved by using the coefficient matrix. The combined effects of elastic boundary conditions, elastic medium, dispersion patterns and nonlocal effects are examined by solving this eigen-value problem constructed with Fourier infinite series. Free vibration frequencies are calculated for carbon nanotube reinforced nanobeam under different rigid or restrained boundary conditions, including an elastic medium Winkler-Pasternak type. A comprehensive parametric study is performed, with the particular focus on the influences of distribution pattern, elastic boundary and medium parameter on the free vibrational response of the composite nanobeam with different graded distributions. It is concluded that the addition of a small amount of carbon nanotube material can reinforce the stiffness of the nanobeam, and its free vibration performance is significantly affected by the distribution patterns, elastic medium and boundary conditions.

Nonlocal stability of curved carbon nanotubes conveying fluid based on Eringen’s nonlocal elasticity theory in a thermomagnetic environment

Nonlocal stability of curved carbon nanotubes conveying fluid based on Eringen’s nonlocal elasticity theory in a thermomagnetic environment

Dislocations in nonlocal simplified strain gradient elasticity: Eringen meets Aifantis

The nonlocal strain gradient elasticity theory is used to address mechanical problems at small scales where size effects and regularization cannot be neglected. In this work, dislocations are investigated in the framework of nonlocal simplified first strain gradient elasticity. It is shown that nonlocal simplified strain gradient elasticity is the unification of the theories of Eringen’s nonlocal elasticity of Helmholtz type and simplified first strain gradient elasticity. Nonlocal simplified strain gradient elasticity contains two characteristic lengths, namely the characteristic length of nonlocal elasticity of Helmholtz type and the characteristic length of simplified first strain gradient elasticity. The advantage of nonlocal simplified first strain gradient elasticity is that the displacement, elastic distortion, plastic distortion, total stress, Cauchy stress and double stress fields of screw and edge dislocations which are calculated here are nonsingular and finite everywhere. Moreover, the Peach-Koehler force of two screw dislocations and two edge dislocations is derived and it is shown that the Peach-Koehler force is also nonsingular. Numerical examples for all dislocation fields of screw and edge dislocations in aluminum are given.

Nonlocal Elasticity Yields Equilibrium Patterns in Phase Separating Systems

Recent experiments demonstrated the emergence of regular mesoscopic patterns when liquid droplets form in an elastic gel after cooling. These patterns appeared via a continuous transition and were smaller in stiffer systems. We capture these observations with a phenomenological equilibrium model describing the density field of the elastic component to account for phase separation. We show that local elasticity theories are insufficient, even if they allow large shear deformations. Instead, we can account for key observations using a nonlocal elasticity theory to capture the gel’s structure. Analytical approximations unveil that the pattern period is determined by the geometric mean between the elastocapillary length and a nonlocality scale. Our theory highlights the importance of nonlocal elasticity in soft matter systems, reveals the mechanism of this mesoscopic pattern, and will improve the engineering of such systems. Published by the American Physical Society 2024

A multimesh finite element method for integral nonlocal elasticity using mesh-decoupling technique

This study presents a generalized multimesh nonlocal finite element method (M2-FEM) that addresses several long-standing challenges in the numerical simulation of Eringen’s integral elasticity theories, often used to model nonlocal effects. The major challenges in the numerical simulation of integral boundary value problems are primarily rooted in the coupling of the spatial discretization of the global (parent) and integral (child) domains, which severely restricts both the applicability and the computational efficiency of existing algorithms by imposing an implicit trade-off between the accuracy of the child domain and the resources required by the overall parent domain. One of the defining contributions of this study consists in the development of a mesh-decoupling technique that generates isolated sets of meshes such that the parent and child domains can be discretized and approximated independently. This mesh-decoupling has multi-fold advantages on the simulation of integral theories such that, when compared to existing state-of-the-art techniques, the proposed algorithm achieves a greater ability to adopt generalized integral kernel functions, the ability to handle non-regular (non-rectangular) domains via unstructured meshing, and simultaneously better numerical accuracy and efficiency (hence allowing greater flexibility in both mesh size and computational cost trade-off decisions). Both benchmark studies and several case studies are conducted to assess the effectiveness of M2-FEM. Simulation results confirm that M2-FEM can accurately simulate integral nonlocal problems and demonstrate its multi-fold computational advantages over existing mesh-coupling-based nonlocal finite element methods. Overall, the proposed M2-FEM algorithm is very general and it can be applied to a variety of integral theories, even beyond nonlocal elasticity.

Elastic medium and torsional spring effects on the nonlocal dynamic of functionally graded porous nanotubes

In this study, Eringen’s nonlocal elasticity theory that applies the small size effects in functionally graded porous nanotubes embedded in an elastic matrix is discussed. The material properties of functionally graded porous nanotubes are taken into account to vary over the radius direction with a rule of mixture. The free torsional vibration relation according to nonlocal elasticity theory, via Hamilton’s principle, is obtained and an eigenvalue solution is constructed for the free torsional vibration response of the presented work. The presented analytical model is validated by comparing the calculated mathematical results for homogeneous nanotubes with rigid and non-rigid boundary conditions. Special attention is given to deformable boundary conditions, porosity coefficient, material grading coefficient and also to the influence of elastic medium on the free torsional vibration frequencies. In this paper, it has been proven that the influence of length, elastic medium, elastic torsional spring rigidities, material grading and porosity coefficients on the vary in the torsional vibration frequency of the functionally graded nanotube is not small.

Two-dimensional nonlocal Eshelby’s inclusion theory: eigenstress-driven formulation and applications

The classical Eshelby’s theory, developed based on local linear elasticity, cannot be applied to inclusion problems that involve nonlocal (long range) elastic effects often observed in micromechanical systems. In this study, we introduce the extension of Eshelby’s inclusion theory to nonlocal elasticity. Starting from Eringen’s integral formulation of nonlocal elasticity, an eigenstress-driven nonlocal Eshelby’s inclusion theory is presented. The eigenstress-driven approach is shown to be a valid mathematical extension of the classical eigenstrain-driven approach in the context of nonlocal inclusion problems. Two individual numerical approaches are developed and applied to simulate inclusion problems and numerically extract the corresponding nonlocal Eshelby tensor. The numerical results obtained from both approaches confirm the validity of the derived nonlocal Eshelby tensor and its ability to capture the non-uniform eigenstress distribution within an elliptic inclusion. These results also help reveal the fundamental difference between the mechanical behaviour of the classical local and the nonlocal inclusion problems. The eigenstress-driven nonlocal inclusion theory could provide the necessary theoretical foundation for the development of homogenization methods of nonlocal heterogeneous media.

Application of the Green's function method for static analysis of nonlocal stress-driven and strain gradient elastic nanobeams

This paper focuses on the static analysis of a generally restrained Euler-Bernoulli nanobeam (GRNB), which is subjected to arbitrary distributed or concentrated loads. The small length scale effect is introduced through a strongly nonlocal integral elastic law called the stress-driven nonlocal Euler-Bernoulli beam model, and a weakly nonlocal elastic law called the strain-gradient Euler-Bernoulli beam model. The stress-driven nonlocal theory and the strain gradient elasticity theory are both employed, to formulate the differential equation governing the Euler-Bernoulli nanobeams. Both theories are ruled by a similar 6th order differential equation, with distinct higher-order boundary conditions. The paper demonstrates that the strain gradient theory can be also reformulated as a stress-driven nonlocal theory with unsymmetrical kernel, in contrast to the initial stress-driven nonlocal theory which has symmetrical exponential kernel. Additionally, some theoretical backgrounds about both nonlocal elasticity theorems are provided. Exact solutions for both nanobeam models are presented, using Green's function method. The parameters of the Green's functions are derived for each beam theory, and then utilized to establish the static displacement function of both nanobeam theories. Exact solutions are provided for the bending of both nanobeams with various end boundary conditions, that account for the small length scale effects. The paper also compares the responses of the stress-driven Euler-Bernoulli beam and the strain gradient Euler-Bernoulli beam, showing close deflections and similar effects of increasing the nonlocal parameter on beam stiffness. A major difference highlighted between the two theories is that the strain gradient theory predicts a homogeneous strain state for homogeneous stress state, as opposed to the stress-driven theory.

Higher-order multi-scale computational approach and its convergence for nonlocal gradient elasticity problems of composite materials

A novel higher-order multi-scale computational approach for efficient nonlocal gradient elasticity simulation of composite materials is proposed in this paper. Based on the use of decoupling technique, raw fourth-order nonlocal gradient equations with periodically oscillatory coefficients are decoupled as the new second-order multi-scale equations for reducing the continuity requirements when implementing multi-scale simulation. Next, a novel macro-micro coupled multi-scale computational model with higher-order corrected terms is rigorously devised for high-resolution multi-scale and nonlocal analysis of composite materials via multi-scale asymptotic analysis. Then, the local error analysis of higher-order multi-scale solutions in the point-wise sense illustrates that establishing higher-order asymptotic corrected terms plays a vital role in investigating the micromechanical mechanisms. Moreover, a rigorous global error estimation with an explicit rate of higher-order multi-scale solutions is first derived in the energy norm sense. In addition, an efficient multi-scale numerical algorithm is presented to effectively simulate nonlocal gradient elasticity problems of composite materials based on finite element method (FEM). And we also derive the convergence estimation of the proposed numerical algorithm. Finally, numerical experiments are conducted to gauge the efficiency and accuracy of the proposed higher-order multi-scale method. This paper offers a unified higher-order two-scale computational framework for enabling nonlocal gradient elasticity behavior simulation of composite materials.

Modeling of sodiation-induced deformation of Sn anode based on the stress-driven nonlocal integral elasticity

Modeling of sodiation-induced deformation of Sn anode based on the stress-driven nonlocal integral elasticity

Numerical investigation of non-local elasticity theory for free vibration of polymer embedded multi-layer graphene sheets

The present study investigates the free vibration behavior of multi-layered graphene sheets with various shapes embedded in a polymer matrix under different boundary conditions based on Eringen non local elasticity. The governing equations are obtained through the minimization of potential energy or Newton’s law which is not only equivalent to Hamilton’s principle and mutually can be derived from each other. Differential Quadrature and pb2 Rayleigh Ritz procedures are applied to Multi-layered Graphene sheets employing Eringen’s elasticity and classical plate theory. Results indicate that the resonant frequencies of the sheets are overestimated up to 60% using the classical elastic model. Depending on the shapes of Graphene sheets, small-scale parameters, boundary conditions, stiffness of polymer matrix, the number of layers and van der Waal’s interaction on the vibrational characteristics of graphene sheets are studied, and the results are tabulated. Conclusions are drawn and recommendations are provided for future work.

In-plane bending vibration of L-shaped cantilever nanobeams carrying a tip nanoparticle by nonlocal elasticity

In-plane bending vibration of L-shaped cantilever nanobeams carrying a tip nanoparticle by nonlocal elasticity

Thermal effects on nonlinear vibration of nonlocal nanobeam embedded in nonlinear elastic medium

Nanotechnology has an impact on our lives in a many ways, from better medical treatments and more efficient energy sources to stronger and lighter materials and advanced electronics and this article presents the implementation of a perturbation method for the vibration analysis of simply supported and clamped–clamped Euler–Bernoulli nanobeams resting on nonlinear elastic foundations in thermal environment using nonlocal elasticity theory. Hamilton's principle is used to construct the differential equation of motion of a nanobeam in conjunction with appropriate boundary conditions. The equations of motion of the Euler–Bernoulli nanobeam are determined using nonlocal elasticity theory. It is shown how thermal loadings affect the vibration of the Euler–Bernoulli nanobeam. The multiple scale method, which is one of the perturbation method, is used to get an approximated solution for the presented system. The effects of temperature, Winkler, Pasternak and nonlinear foundation parameters on the vibration analysis of simply supported and clamped–clamped nanobeams are determined and results are given in tables and graphs.