Nonlocal Finite Element Method Research Articles

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.

An eigenvalue solution for nonlocal vibration of guide supported perfect/imperfect functionally graded power‐law and sigmoid nanobeams on one‐parameter elastic foundation

AbstractTo the best of the authors' knowledge, this study for the first time, proposes a finite element (FE) model to investigate the size‐dependent vibrational responses of guide supported imperfect functionally graded (FG) nonlocal beams embedded in an elastic foundation in the context of nonlocal elasticity theory (NET). The main objective of this paper is to present a nonlocal FE solution for vibration analysis of imperfect FG nanobeams including nonlocal effect, power‐law distribution function, sigmoid distribution function, even porosity model, uneven porosity model, elastic foundation parameter, and guide support condition. It is considered that perfect/imperfect FG nanobeams are made of silicon carbide (SiC) and aluminum (Al) and that beam properties vary along the height direction. In addition, the nanobeams rest on a one‐parameter elastic foundation and this foundation provides the interaction between the structure and the soil with elastic springs. The frequency values of perfect/imperfect FG power‐law and sigmoid nanobeams obtained by using a nonlocal FE method are shown via tables and figures.

A new eigenvalue problem solver for thermo‐mechanical vibration of Timoshenko nanobeams by an innovative nonlocal finite element method

In this study, size‐dependent thermo‐mechanical vibration analysis of nanobeams is examined. Size‐dependent dynamic equations are obtained by implementing Hamilton's principle based on Timoshenko beam theory and then combined with stress equation of nonlocal elasticity theory. The separation of variables total method and finite element formulation is utilized to solve the eigenvalue problem. Local and nonlocal stiffness and mass matrices are firstly derived by using a weighted residual method for the finite element analysis. The accuracy of the finite element solution is demonstrated by comparisons with the earlier studies. Then, nondimensional frequencies of nanobeams with different boundary conditions based on a nonlocal finite element method are presented for vibration analysis that cannot be analytically solved under different parameters. It is aimed to emphasize the importance of the nonlocal finite element method in the size‐dependent vibration behavior of nanobeams which form different components of nano‐electro‐mechanical systems.

Thermal Vibration of Zinc Oxide Nanowires by using Nonlocal Finite Element Method

Zinc oxide nanowires (ZnO NWs) can be used in some NEMS applications due to their remarkable chemical, physical, mechanical and thermal resistance properties. In terms of the suitability of such NEMS organizations, a correct mechanical model and design of ZnO NWs should also be established under different effects. In this study, thermal vibration analyses of elastic beam models of ZnO NWs are examined based on Eringen's nonlocal elasticity theory. The resulting equation of motion is solved with a finite element formulation developed for the atomic size-effect and thermal environment. The vibration frequencies of ZnO NWs with different boundary conditions are calculated under nonlocal parameter and temperature change values ​​and numerical results were discussed.

Free vibration characteristics of nonlocal viscoelastic nano‐scaled plates with rectangular cutout and surface effects

AbstractIn the present study, the viscoelastic free vibration behavior of nano‐scaled plates is studied by employing a finite element method based on the two‐phase nonlocal integral theory. Various boundary conditions, surface effects and cutouts have been assumed. The principle of total potential energy is used for developing the nonlocal finite element method, and the classical plate theory is assumed to derive the formulations. By numerically solving the eigenvalue problem, which has been obtained by the variational principle, the complex eigenvalues of free vibration of the viscoelastic nano‐scaled plates are acquired. The current results are compared with those available in the literature and those obtained by commercial finite element software, and the influences of the nonlocal parameter, viscoelastic parameter, geometrical parameters (e.g. cutout and size), surface effects and different boundary conditions on the complex eigenvalues are studied. It is noted that the current method is able to handle quite versatile boundary conditions and geometries like cutouts, which are rather difficult (or impossible) to be tackled by employing other methods available in most researches.

Stability analysis of nanobeams placed in electromagnetic field using a finite element method

In this work, the buckling analysis of the nanobeams placed in electromagnetic field is presented via a nonlocal finite element method based on Eringen’s nonlocal elasticity theory. The governing differential equation is derived by implementing minimum total potential energy principle. A finite element method is proposed for solution of nonlocal buckling of nanobeam placed in electromagnetic field. The contribution of this article is the use of interpolation functions and the nonlocal elasticity theory to form the stiffness matrices and geometric stiffness matrices of the electromagnetic nanobeam for buckling analysis. A detailed study is performed to indicate the influences of some parameters such as Hartmann parameter (Ha), mode number (n), nonlocal parameter (e0a), length of nanobeam (L), and boundary conditions on the buckling loads of Euler-Bernoulli nanobeams. The values of buckling load of electromagnetic nanobeams are obtained via a finite element solution, examined through several numerical examples, and demonstrated by tables and graphs.

Vibration energy flow analysis of periodic nanoplate structures under thermal load using fourth-order strain gradient theory

A nonlocal Kirchhoff plate model with fourth-order strain gradient theory is firstly proposed to study variations with band gap frequencies and vibration energy distribution. The temperature rise is supposed to vary linearly through the thickness of the periodic nanoplate structures. The dynamic equations of finite periodic nanoplate structures under thermal load with the small-scale effect and the nonlinear membrane strain taken into consideration are derived based on the finite element method. The structural intensity approach is developed to predict the vibration energy flow of the periodic nanoplates. Effects of the temperature rise on band gaps and the structural intensity are divided into two parts. One is nonlinear effects of temperature rise, and the other is effects of the thermal load. In the numerical calculation, the natural frequencies of single-layer graphene sheets computed by the nonlocal finite element method with fourth-order strain gradient agree well with analytical results, which validate the effectiveness of the present method. The influences of nonlocal parameters and thermal load on the band gap and structural intensity are considered, respectively. The boundary value has been achieved to determine the critical mechanical load or thermal load for analyzing in depth the effects of the mechanical load and thermal load on the structural intensity. The proposed method shows that the vibration energy flow pattern may be controlled by adjusting the magnitude of the mechanical or thermal load.

Size-dependent transverse and longitudinal vibrations of embedded carbon and silica carbide nanotubes by nonlocal finite element method

In this study, free vibration analyses of embedded carbon and silica carbide nanotubes lying on an elastic matrix are performed based on Eringen’s nonlocal elasticity theory. These nanotubes are modeled as nanobeam and nanorod. Elastic matrix is considered as Winkler–Pasternak elastic foundation and axial elastic media for beam and rod models, respectively. The vibration formulations of the beam and rod are derived by utilizing Hamilton’s principle. The obtained equations of motions are solved by the method of separation of variables and finite element-based Hermite polynomials for various boundary conditions. The effects of boundary conditions, system modeling, structural sizes such as length, cross-sectional sizes, elastic matrix, mode number, and nonlocal parameters on the natural frequencies of these nanostructures are discussed in detail. Moreover, the availability of size-dependent finite element formulation is investigated in the vibration problem of nanobeams/rods resting on an elastic matrix.

Features of the software implementation of the numerical solution of stationary heat equation taking into account the effects of nonlocal finite element method

The features of the numerical solution of the two-dimensional equation of stationary heat conduction are considered taking into account nonlocal effects. The finite element method using isoparametric finite elements was chosen as the numerical solution scheme. A method for approximating the zone of nonlocal influence was proposed, and an estimate of the time for calculating the coefficients of the thermal conductivity matrix was also constructed. The influence of the main parameters of the model on the final results is studied, and a polynomial family of functions of non-local influence is also studied. The results are compared with the results obtained with the classical approach.

Nonlocal vibration analysis of Ti-6Al-4V/ZrO2 functionally graded nanobeam on elastic matrix

In the present study, a nonlocal finite element method (FEM) is proposed to investigate the free vibration of functionally graded (FG) nanobeams resting on two parameters, the Winkler–Pasternak elastic foundation. Using the Eringen’s nonlocal elasticity theory, the Euler–Bernoulli beam model is implemented. The equations of motion are obtained by using Hamilton’s principle. Material properties of the beam vary in the thickness (height) direction based on the power law. The frequencies of functionally graded nanobeam are obtained for simply supported (S-S) boundary conditions with various values of power law exponent, small-scale (nonlocal) parameter, Winkler foundation parameter, and Pasternak foundation parameter. Vibration response of nano-scaled functionally graded beam resting on the Winkler–Pasternak elastic foundation is investigated via the nonlocal finite element method. A comparison of the numerical results of the present study with those from the open literature demonstrates a good agreement. Also, the difference between classical elasticity theory and nonlocal elasticity theory is discussed in this study.

On the torsional vibration of nanorods surrounded by elastic matrix via nonlocal FEM

Nonlocal dynamic torsional response of nanorods embedded in elastic media is investigated. It is considered that mechanical behavior of elastic media is supposed to be like linear foundation model. The nonlocal dynamic torsion equation is obtained according to Hamilton's Principle. Application of solved motion of equation is performed for nanorod models that have torsional spring attachment at the one end as well as three different general boundary conditions. Moreover, the formulation of nonlocal finite element method (NL–FEM) based on weighted residual that considers stiffness of elastic media and attachment ratio is attained; this finite element formula is new in the literature. The nondimensional torsional frequencies are presented under nanorod length, nondimensional nonlocal parameter, slenderness ratio, nondimensional media stiffness parameter and stiffness ratio of attachment as tables and graphics comparatively with NL–FEM. This study is exhibited that NL–FEM can be used for torsional vibration analysis of nanorods embedded in elastic media.

A Size-Dependent Coupled Symplectic and Finite Element Method for Steady-State Forced Vibration of Built-Up Nanobeam Systems

A novel size-dependent coupled symplectic and finite element method (FEM) is proposed to study the steady-state forced vibration of built-up nanobeam system resting on elastic foundations. The overall system is modeled as a combination of nonlocal Timoshenko beams. A new analytical subsystem modeling with formulation and another numerical subsystem modeling are developed and discussed. In the analytical subsystem model, the uniform nanobeams are modeled and solved by a new approach based on a series of analytical symplectic eigensolutions. The numerical subsystem model applies a nonlocal FEM to solve nonuniform nanobeams. Analytical and numerical solutions are presented, and a proper comparison between the two approaches is established. Comprehensive and accurate numerical result is subsequently presented to illustrate the accuracy and reliability of the coupled method. The new results established are expected to have reference values for future studies.

Free vibration analysis Silicon nanowires surrounded by elastic matrix by nonlocal finite element method

Higher-order theories are very important to investigate the mechanical properties and behaviors of nanoscale structures. In this study, a free vibration behavior of SiNW resting on elastic foundation is investigated via Eringen\'s nonlocal elasticity theory. Silicon Nanowire (SiNW) is modeled as simply supported both ends and clamped-free Euler-Bernoulli beam. Pasternak two-parameter elastic foundation model is used as foundation. Finite element formulation is obtained nonlocal Euler-Bernoulli beam theory. First, shape function of the Euler-Bernoulli beam is gained and then Galerkin weighted residual method is applied to the governing equations to obtain the stiffness and mass matrices including the foundation parameters and small scale parameter. Frequency values of SiNW is examined according to foundation and small scale parameters and the results are given by tables and graphs. The effects of small scale parameter, boundary conditions, foundation parameters on frequencies are investigated.

Free Vibration Analysis of BNNT with Different Cross-Sections via Nonlocal FEM

In the present study, free vibration behaviors of of carbon nanotube (CNT) and boron nitride nanotube (BNNT) have been investigated via Eringen’s nonlocal continuum theory. Size effect has been considered via nonlocal continuum theory. Nanotubes have become popular in the world of science thanks to their characteristic properties. In this study, free vibrations of Boron Nitride Nanotube (BNNT) and Carbon Nanotube (CNT) are calculated using the Nonlocal Elasticity Theory. Frequency values are found via both analytical and finite element method (FEM). Galerkin weighted residual method is used to obtain the finite element equations. BNNT and CNT are modeled as Euler - Bernoulli Beam and solutions are gained by using four different cross-section geometries with three boundary conditions. Selected geometries are circle, rectangle, triangle, and square. Frequency values are given in tables and graphs. The effect of cross-section, boundary conditions and length scale parameter on frequencies has been investigated in detail for BNNT.

A non-local plasticity model of stimulated volume evolution during hydraulic fracturing

Hydraulic fracturing in naturally fractured rocks often leads to the creation of a stimulated zone in which the target rock formation is deformed and fractured by the reactivation and shear dilation of natural fractures and the plastic deformation, damaging, and fracturing of the bulk. In this paper, we present a novel mathematical model with the goal of simulating the evolution of the stimulated volume during hydraulic fracturing. This was achieved by introducing an equivalent continuum non-local poro-elastic-plastic zone of enhanced permeability for the stimulated region, characterized by an internal length scale. The non-local plastic constitutive behavior of the rock, combined with the classical Biot’s poroelastic theory, was implemented using a new implicit C0 non-local finite element method. A predictor-corrector return algorithm for the non-local plasticity model was formulated as an extension of the classical plasticity algorithm. To improve the performance of the iterative solution scheme, a consistent algorithmic stiffness tangent modulus was developed. First, the elastic-plastic constitutive behavior of the proposed methodology is verified using the standard non-porous biaxial compression test with strain softening behavior. Next, it is verified that the poro-elastic-plastic model correctly simulates the evolution of the stimulated zone and the subsequent change in the flow and fluid pressure for several hydraulic fracturing examples under various far-field in-situ stress conditions. Lastly, the non-local poro-elastic-plastic model is shown to be mesh-independent and capable of capturing a wide range of complex fracturing behavior.

Stress evaluation in displacement-based 2D nonlocal finite element method

AbstractThe evaluation of the stress field within a nonlocal version of the displacement-based finite element method is addressed. With the aid of two numerical examples it is shown as some spurious oscillations of the computed nonlocal stresses arise at sections (or zones) of macroscopic inhomogeneity of the examined structures. It is also shown how the above drawback, which renders the stress numerical solution unreliable, can be viewed as the so-called locking in FEM, a subject debated in the early seventies. It is proved that a well known remedy for locking, i.e. the reduced integration technique, can be successfully applied also in the nonlocal elasticity context.

Investigations on structural intensity in nanoplates with thermal load

The finite element method (FEM) based on the nonlocal Kirchhoff plate theory with second order strain gradient is developed to derive the dynamic equations of nanoplate under thermal load with the small scale effect taken into consideration. The characteristics of transmission and distribution of the steady-state energy flows in the rectangular nanoplate are analyzed based on the structural intensity approach. In the numerical calculation, the natural frequencies of single layer graphene sheets (SLGS) computed by nonlocal FEM agree well with theoretical results of nonlocal strain gradient plate theory, which validate the reliability of the present method. The effects of nonlocal parameters, mechanical load and thermal load on structural intensity are considered. It can be found that the small scale effect is not same for different applying positions and excitation frequencies. The influence of mechanical load on vibration energy flow paths may be cancelled by thermal load, and the effect of thermal load on vibration energy flow paths also may be cancelled by mechanical load. The critical thermal load may be found to determine whether thermal load play a more important role in form energy flow of SLGS than mechanical load.

On the Nonexistence of a Feasible Solution in the Context of the Differential Form of Eringen’s Constitutive Model: A Proposed Iterative Model Based on a Residual Nonlocality Formulation

The motivation of this paper is to highlight and discuss critically the details of two main aspects related to Eringen’s nonlocal constitutive model. The first aspect is to point out the inconsistency of the integral and differential forms of Eringen’s nonlocal constitutive model. In addition, to point out the ill-posed form and physical infeasibility of the results which may be obtained by using the differential form of the model. The critical analysis focuses on the lack of consistency between the set of the boundary constraints required by the differential form of Eringen’s model and the set of the prescribed boundary conditions of the nonlocal static equilibrium problem. Because of this lack of consistency between the two constitutive forms, it can be concluded that the formulation in context of the differential form is ill-posed and the existence of a feasible solution is questionable and might not be admitted. The second aspect deals with the intractability of the analytical solution of the nonlocal continuum problems based on the integral form of Eringen’s nonlocal constitutive model (IENCM). In the meantime, it mentions the cumbersome of numerical work required by the direct computational methods such as the nonlocal finite element method. The complexity of using the integral form of Eringen’s constitutive model and the lack of existence of a feasible solution by using the differential form of Eringen’s constitutive model lead to the mandatory need for developing an efficient iterative computational approach based on the integral form of Eringen’s constitutive model. In this paper, an iterative computational method, based on the nonlocality residual formulation for nonlocal continuums, capable to investigate different elasticity problems, is proposed. The traditional local continuum solution is taken as an initial solution. To point out the inconsistences between the integral and differential forms of Eringen’s nonlocal constitutive model, and to illustrate the efficiency and capability of the proposed iterative model, four static bending of beam problems with different nature are solved.

Bending, buckling and free vibration analysis of Euler-Bernoulli nanobeams using Eringen’s nonlocal integral model via finite element method

In the present study, finite element formulations are derived for static bending, linear buckling and free vibration analysis of nanobeam structures by utilizing the integral form of Eringen nonlocal model. Formulations are developed according to the minimum total potential energy principle by presenting the differentiation operations explicitly. As being distinct from other studies, a non-uniform mesh distribution is proposed for the corresponding analytical expressions of the beam deflections. With this aid, the discontinuous nature of rotation angle, which is encountered at boundaries of the beam, is aimed to be captured. Many numerical examples are solved, and compared with the exact solutions reported in the literature to demonstrate the versatility of the non-local finite element method with the proposed mesh configuration. It is found out that, with the suggested mesh distribution, the number of elements can be decreased dramatically without sacrificing from the accuracy, which consequently leads to a considerable reduction in the computational cost.

Nonlocal elasticity in plates using novel trial functions

This study presents the Ritz formulation, which is based on boundary characteristic orthogonal polynomials (BCOPs), for the two-phase integro-differential form of the Eringen's nonlocal elasticity model. This approach is named the nonlocal Ritz method (NL-RM). This feature greatly reduces the computational cost compared to the nonlocal finite-element method (NL-FEM). Another advantage of this approach is that, unlike NL-FEM, the nonlocal mass and stiffness matrices are independent of the mesh distribution. Here, these formulations are applied to study the static-bending and free-dynamic analyses of the Kirchhoff plate model. In this paper, novel 2D BCOPs of the plate are derived as coordinate functions. These polynomials are generated using a modified Gram-Schmidt process and satisfy the given geometrical boundary conditions as well as the natural boundary conditions. The accuracy and convergence of the presented model, demonstrated through several numerical examples, are discussed. A concise argument on the advantages of NL-RM compared to NL-FEM is also provided.