Nonlocal Continuum Model Research Articles

Discrete and non-local elastica

In this paper, the buckling and post-buckling behavior of an elastic lattice system referred to as the discrete elastica problem is investigated using an equivalent non-local continuum approach. The geometrically exact post-buckling analysis of the elastic chain, also called Hencky system, is first numerically solved using the shooting method. This discrete physical model is also mathematically equivalent to a finite difference formulation of the continuum elastica. Starting from the exact difference equations of the discrete problem, a continualization method is applied for approximating the difference operators by differential ones, in order to better characterize the discrete system by an enriched continuous one. It is shown that the new continuum associated with the discrete system exactly fits the discrete elastica post-buckling problem, where the non-locality is of Eringen׳s type (also called stress gradient non-local model). An asymptotic expansion is performed for both the discrete and the non-local continuum models, in order to approximate the post-buckling branches of the discrete system. Some numerical investigations show the efficiency of the non-local approach, especially for capturing the scale effects inherent to the cell size of the lattice model.

Nonlocal and shear effects on column buckling of single-layered membranes from stocky single-walled carbon nanotubes

Axial buckling behavior of single-layered membranes from vertically aligned single-walled carbon nanotubes is studied in the context of the nonlocal continuum theory of Eringen. To this end, useful discrete models based on the nonlocal Rayleigh, Timoshenko, and higher-order beam theories are developed to evaluate critical buckling loads associated with both in-plane and out-of-plane buckling modes. In discrete models, the size of the eigenvalue equations to be solved drastically magnifies for highly populated membranes. Thereby, development of models whose computational efforts do not affected by the population of the membrane is of great advantageous. To bridge this scientific gap, appropriate nonlocal continuous models are established based on the developed discrete models. The accuracy of the proposed discrete and continuous models is checked and remarkable results are achieved. Subsequently, the roles of the influential factors on both in-plane and out-of-plane axial buckling loads are addressed. The obtained results can be regarded as a basic step in examining of axial buckling mechanisms of more complex systems consist of multi-layered membranes from parallel or even orthogonal single-walled carbon nanotubes.

Orthotropic patterns of Pasternak foundation in smart vibration analysis of magnetostrictive nanoplate

In this research, free vibration of rectangular nanoplate made of magnetostrictive materials is studied while it is focused on elastic medium as an efficient stability factor. For this purpose, Pasternak foundation is developed by considering orthotropy angle where the effect of Pasternak shear modulus is investigated in different directions. Since the nanoplate is subjected to the coil, a feedback control system follows the effects of uniform magnetic field on vibration characteristics of magnetostrictive nanoplate. So, Reddy’s third-order shear deformation theory along with Eringen’s nonlocal continuum model are utilized in order to derive motion equations at nanoscale using Hamilton’s principle. Five coupled motion equations solved by differential quadrature method in two-dimensional space by considering different boundary conditions. Results indicate that with appropriative selection for orthotropy angle, normal, and shear Pasternak foundation modulus, it is possible to achieve optimal and desire values to more stability of magnetostrictive nanoplate. These findings can be used in automotive industry, communications equipment in nano- and microstructures.

Nonlocal continuum model for vibration of single-layered graphene sheets based on the element-free kp-Ritz method

In this paper, an implementation of the kp-Ritz method with the nonlocal continuum model as an element-free computational framework has been performed to investigate the free vibration behavior of a single-layered graphene sheet (SLGS). The nonlocal continuum model, which combines the Eringen nonlocal constitutive equation with the classical plate theory, has the ability to take the small scale effect into account. The study has shown that the element-free kp-Ritz method is an efficient approach for solving nonlocal continuum model for the SLGS vibration. The accuracy of the kp-Ritz method has been validated through comparison with the MD computation. The values of nonlocal parameter used in the study have been derived by matching the frequency of nonlocal continuum model with that of the MD model for different sizes and boundary conditions. The simulated results have illustrated that the value of nonlocal parameter depends on the sizes and boundary conditions of SLGS.

Bending and buckling analyses of functionally graded material (FGM) size-dependent nanoscale beams including the thickness stretching effect

This paper addresses theoretically the bending and buckling behaviors of size-dependent nanobeams made of functionally graded materials (FGMs) including the thickness stretching effect. The size-dependent FGM nanobeam is investigated on the basis of the nonlocal continuum model. The nonlocal elastic behavior is described by the differential constitutive model of Eringen, which enables the present model to become effective in the analysis and design of nanostructures. The present model incorporates the length scale parameter (nonlocal parameter) which can capture the small scale effect, and furthermore accounts for both shear deformation and thickness stretching effects by virtue of a sinusoidal variation of all displacements through the thickness without using shear correction factor. The material properties of FGM nanobeams are assumed to vary through the thickness according to a power law. The governing equations and the related boundary conditions are derived using the principal of minimum total potential energy. A Navier-type solution is developed for simply-supported boundary conditions, and exact expressions are proposed for the deflections and the buckling load. The effects of nonlocal parameter, aspect ratio and various material compositions on the static and stability responses of the FGM nanobeam are discussed in detail. The study is relevant to nanotechnology deployment in for example aircraft structures.

Exact solutions of bending deflections for nano-beams and nano-plates based on nonlocal elasticity theory

In this paper, we applied the nonlocal continuum mechanics to deriving a complete and asymptotic representation of the infinite higher-order governing differential equations for nano-beam and nano-plate models. Results show that the vanishing issue of small scale effects under some boundary conditions such as a clamped nano-beam subjected to distributed uniformly load is solved by using the present nonlocal continuum model. Several typical examples for nano-beam and nano-plate are studied and the scale effect on bending deflection is discussed at length. The analytical prediction for nonlocal bending deflection gives a good prediction of molecular dynamics simulation results which confirms the accuracy of the present model. It is found that when the external characteristic length tends to internal characteristic length, a property of a rigid body for the nano-beam and nano-plate is presented.

Lattice fractional calculus

Integration and differentiation of non-integer orders for N-dimensional physical lattices with long-range particle interactions are suggested. The proposed lattice fractional derivatives and integrals are represented by kernels of lattice long-range interactions, such that their Fourier series transformations have a power-law form with respect to components of wave vector. Continuous limits for these lattice fractional derivatives and integrals give the continuum derivatives and integrals of non-integer orders with respect to coordinates. Lattice analogs of fractional differential equations that include suggested lattice differential and integral operators can serve as an important element of microscopic approach to nonlocal continuum models in mechanics and physics.

Plane strain and plane stress elasticity under fractional continuum mechanics

In this paper, the application of non-local fractional continuum model for plane strain and plane stress elasticity is presented. The kinematics and stress concepts are discussed, and governing equations in terms of displacements for both plane problems are defined. The numerical implementation utilising generalised finite difference method is shown in detail. Three cases are solved to indicate the role of order of fractional continua and length scale: biaxial tension, pure shear and complex state. Classical (local) solution is obtained as a special case.

Axisymmetric vibration of SWCNTs in water with arbitrary chirality based on nonlocal anisotropic shell model

The axisymmetric vibration of three kinds of zigzag, armchair and chiral single-walled carbon nanotubes (SWCNTs) in water is studied in the paper based on the nonlocal anisotropic shell model. The present model is justified compared with the existing experimental observation and available data in literature. It is shown that for the tubes with the same diameter, the radial breathing frequencies from lower to upper limits are zigzag, chiral and armchair tubes, respectively. However, the difference in frequency diminishes with the increase of diameter. Furthermore, we found that the axisymmetric radial, longitudinal and torsional vibrational frequencies of SWCNTs are mainly sensitive to the values of radius and the scale coefficient, but cannot be substantially changed by their atomic structure. Moreover, we can get that the local shell model for CNT analysis would lead to an overprediction of the vibrational frequencies of almost all modes. So the nonlocal isotropic continuum models are practical in analyzing CNTs of large geometric sizes. Finally, the results show that the effects of fluid motion surrounding CNTs can be ignored for the large wave numbers.

Analytical solutions for bending of transversely or axially FG nonlocal beams

This paper presents the analytical solutions for the size-dependent static analysis of the functionally graded (FG) beams with various boundary conditions based on the nonlocal continuum model. The nonlocal behavior is described by the differential constitutive model of Eringen, which enables to this model to become effective in the analysis and design of nanostructures. The elastic modulus of beam is assumed to vary through the thickness or longitudinal directions according to the power law. The governing equations are derived by using the nonlocal continuum theory incorporated with Euler-Bernoulli beam theory. The explicit solutions are derived for the static behavior of the transversely or axially FG beams with various boundary conditions. The verification of the model is obtained by comparing the current results with previously published works and a good agreement is observed. Numerical results are presented to show the significance of the nonlocal effect, the material distribution profile, the boundary conditions, and the length of beams on the bending behavior of nonlocal FG beams.

Simulation of liquid crystal elastomers using Chebyshev spectral method with a new preconditioner

Liquid crystal elastomers (LCEs) are soft complex materials of potential technological importance because of their remarkable responsiveness to excitations. In the previous work (Zhu et al. Phys. Rev. E 83, 051703 2011), we proposed a non-local continuum model to explore the dynamical behaviors of LCEs. The preliminary simulation demonstrated that the model can successfully capture the shape changing phenomena and other features of LCEs that were observed in real experiments (Camacho-Lopez et al. Nat. Mat. 3, 307---310 2004). However, due to the complexity of the governing equations, especially the velocity equation, the simulation is very time-consuming and thus efficient methods are imperatively needed. In this work, we propose a novel preconditioner for solving the velocity equation using Chebyshev spectral collocation method. Different from the well-known finite difference preconditioning (Orszag J. Comput. Phys. 37, 70---92 1980), the proposed preconditioner is constructed directly from the coefficient matrix of solving the velocity equation, without resorting to other operators such as the finite difference operator. With this preconditioner, very few inner iterations are needed when solving the resulting large scale linear system using GMRES method (Saad and Schultz SIAM J. Sci. STAT. Comput. 7(3), 856---869 1986). The experiments validate the efficiency of the proposed preconditioner.

Small-Scale Effect on Longitudinal Wave Propagation in Laser-Excited Plates

Longitudinal wave propagation in an elastic isotopic laser-excited solid plate with atomic defect (vacancies, interstitials) generation is studied by the nonlocal continuum model. The nonlocal differential constitutive equations of Eringen are used in the formulations. The coupled governing equations for the dynamic of elastic displacement and atomic defect concentration fields are obtained. The frequency equations for the symmetrical and antisymmetrical motions of the plate are found and discussed. Explicit expressions for different characteristics of waves like phase velocity and attenuation (amplification) coefficients are derived. It is shown that coupling between the displacement and defect concentration fields affects the wave dispersion characteristics in the nonlocal elasticity. The dispersion curves of the elastic-diffusion instability are investigated for different pump parameters and larger wave numbers.

Particle methods for pedestrian flow models: From microscopic to nonlocal continuum models

A hierarchy of models for pedestrian flow is numerically investigated using particle methods. It includes microscopic models based on interacting particle system coupled to an eikonal equation, hydrodynamic models using equations for density and mean velocity, nonlocal continuum equations for the density and diffusive Hughes equations. Particle methods are used on all levels of the hierarchy. Numerical test cases are investigated by comparing the above models.

Nonlocal discrete and continuous modeling of free vibration of stocky ensembles of vertically aligned single-walled carbon nanotubes

Free dynamic analysis of transverse motion of vertically aligned stocky ensembles of single-walled carbon nanotubes is of particular interest. A linear model is developed to take into account the van der Waals forces between adjacent SWCNTs because of their bidirectional transverse displacements. Using Hamilton's principle, the discrete equations of motion of free vibration of the nanostructure are obtained based on the nonlocal Rayleigh, Timoshenko, and higher-order beam theories. The application of such discrete models for frequency analysis of highly populated ensembles would be associated with so much computational effort. To overcome such a problem, some useful nonlocal continuous models are established. The obtained results reveal that the newly developed models can successfully capture the predicted fundamental frequencies of the discrete models. Through various numerical studies, the roles of slenderness ratio, radius of the SWCNT, small-scale parameter, population of the ensemble, and intertube distance on the fundamental flexural frequency of the nanostructure are examined and discussed. The capabilities of the proposed nonlocal continuous models in predicting flexural frequencies of the nanostructure are also addressed.

In- and out-of-plane dynamic flexural behaviors of two-dimensional ensembles of vertically aligned single-walled carbon nanotubes

Useful nonlocal discrete and continuous models are developed to explain free vibration of two-dimensional (2D) ensembles of single-walled carbon nanotubes (SWCNTs) in bending. For this purpose, the models are constructed based on the nonlocal Rayleigh, Timoshenko, and higher-order beam theories. In contrast to an individual SWCNT exhibits identical bending behavior in different directions, for 2D ensemble networks of SWCNTs, it is shown that such a fact is completely dissimilar. Such an important issue leads to the definition of in-plane and out-of-plane flexural behaviors for such nanostructures. Subsequently, their corresponding fundamental frequencies are evaluated based on the proposed nonlocal models. The capabilities of the proposed nonlocal continuous models in predicting flexural frequencies of SWCNTs׳ ensembles with different numbers of SWCNTs as well as various levels of slenderness ratios are then explained. Such investigations confirm the high efficiency of the proposed continuous models. This matter would be of great importance in vibration analysis of highly populated ensembles of SWCNTs in which the discrete models may suffer from the size of the governing equations. The roles of the number of SWCNTs, slenderness ratio, intertube distance, small-scale parameter, and radius of the SWCNT on both in-plane and out-of-plane fundamental frequencies are addressed.

The morphing method as a flexible tool for adaptive local/non-local simulation of static fracture

We introduce a framework that adapts local and non-local continuum models to simulate static fracture problems. Non-local models based on the peridynamic theory are promising for the simulation of fracture, as they allow discontinuities in the displacement field. However, they remain computationally expensive. As an alternative, we develop an adaptive coupling technique based on the morphing method to restrict the non-local model adaptively during the evolution of the fracture. The rest of the structure is described by local continuum mechanics. We conduct all simulations in three dimensions, using the relevant discretization scheme in each domain, i.e., the discontinuous Galerkin finite element method in the peridynamic domain and the continuous finite element method in the local continuum mechanics domain.

Torsional vibration of carbon nanotubes: Comparison of two nonlocal models and a semi-continuum model

Abstract In recent years, people were puzzled about two reverse nonlocal models in studying transverse bending of nanobeams. Following the ideologies of both nonlocal models, two kinds of torsional models were constructed to investigate the nonlocal torsional vibration of carbon nanotubes, respectively. Just like the transverse bending of nanobeams, it is strange to observe two opposite size-dependent performances. The first nonlocal continuum model (weakened model) was based on equilibrium equations and nonlocal torsional shear stress relation. Natural frequency decreases with an increase in nonlocal nanoscale parameter, or it increases with increasing length of the carbon nanotube. Thus the torsional stiffness of carbon nanotubes is weakened. On the other hand, the second nonlocal model (enhanced model) was developed from the strain energy variational principle. Natural frequency increases (or decreases) with increasing nonlocal nanoscale parameter (or length of the carbon nanotube), or the nanostructural stiffness is strengthened. For judgment, a torsional semi-continuum model with discrete atomic layers in the cross section of a carbon nanotube was proposed. The relaxation effects on surface atoms were considered in the torsional semi-continuum model. It is concluded that the relaxation type (attractive or repulsive) of surface atoms results in two different nonlocal results. Consequently, both the existing reverse models are proved to be valid.

Nonlocal continuous models for forced vibration analysis of two- and three-dimensional ensembles of single-walled carbon nanotubes

Novel nonlocal discrete and continuous models are proposed for dynamic analysis of two- and three-dimensional ensembles of single-walled carbon nanotubes (SWCNTs). The generated extra van der Waals forces between adjacent SWCNTs due to their lateral motions are evaluated via Lennard-Jones potential function. Using a nonlocal Rayleigh beam model, the discrete and continuous models are developed for both two- and three-dimensional ensembles of SWCNTs acted upon by transverse dynamic loads. The capabilities of the proposed continuous models in capturing the vibration behavior of SWCNTs ensembles are then examined through various numerical simulations. A reasonably good agreement between the results of the continuous models and those of the discrete ones is also reported. The effects of the applied load frequency, intertube spaces, and small-scale parameter on the transverse dynamic responses of both two- and three-dimensional ensembles of SWCNTs are explained. The proposed continuous models would be very useful for dynamic analyses of large populated ensembles of SWCNTs whose discrete models suffer from both computational efforts and labor costs.

Nonlocal continuum-based modeling of breathing mode of nanowires including surface stress and surface inertia effects

Nonlocal and surface effects significantly influence the mechanical response of nanomaterials and nanostructures. In this work, the breathing mode of a circular nanowire is studied on the basis of the nonlocal continuum model. Both the surface elastic properties and surface inertia effect are included. Nanowires can be modeled as long cylindrical solid objects. The classical model is reformulated using the nonlocal differential constitutive relations of Eringen and Gurtin–Murdoch surface continuum elasticity formalism. A new frequency equation for the breathing mode of nanowires, including small scale effect, surface stress and surface inertia is presented by employing the Bessel functions. Numerical results are computed, and are compared to confirm the validity and accuracy of the proposed method. Furthermore, the model is used to elucidate the effect of nonlocal parameter, the surface stress, the surface inertia and the nanowire orientation on the breathing mode of several types of nanowires with size ranging from 0.5 to 4nm. Our results reveal that the combined surface and small scale effects are significant for nanowires with diameter smaller than 4nm.

Peridynamic State-Based Models and the Embedded-Atom Model

AbstractWe investigate connections between nonlocal continuum models and molecular dynamics. A continuous upscaling of molecular dynamics models of the form of the embedded-atom model is presented, providing means for simulating molecular dynamics systems at greatly reduced cost. Results are presented for structured and structureless material models, supported by computational experiments. The nonlocal continuum models are shown to be instances of the state-based peridynamics theory. Connections relating multibody peridynamic models and upscaled nonlocal continuum models are derived.