Nonlocal Constitutive Relations Research Articles

Coupled responses of thermomechanical waves in functionally graded viscoelastic nanobeams via thermoelastic heat conduction model including Atangana–Baleanu fractional derivative

Accurately characterizing the thermomechanical parameters of nanoscale systems is essential for understanding their performance and building innovative nanoscale technologies due to their distinct behaviours. Fractional thermal transport models are commonly utilized to correctly depict the heat transfer that occurs in these nanoscale systems. The current study presents a novel mathematical thermoelastic model that incorporates a new fractional differential constitutive equation for heat conduction. This heat equation is useful for understanding the effects of thermal memory. An application of a fractional-time Atangana–Baleanu (AB) derivative with a local and non-singular kernel was utilized in the process of developing the mathematical model that was suggested. To deal with effects that depend on size, nonlocal constitutive relations are introduced. Furthermore, in order to take into consideration, the viscoelastic behaviour of the material at the nanoscale, the fractional Kelvin–Voigt model is utilized. The proposed model is highly effective in properly depicting the unusual thermal conductivity phenomena often found in nanoscale devices. The study also considered the mechanical deformation, temperature variations, and viscoelastic characteristics of the functionally graded (FG) nanostructured beams. The consideration was made that the material characteristics exhibit heterogeneity and continuous variation across the thickness of the beam as the nanobeam transitions from a ceramic composition in the lower region to a metallic composition in the upper region. The complicated thermomechanical features of simply supported viscoelastic nanobeams that were exposed to harmonic heat flow were determined by the application of the model that was constructed. Heterogeneity, nonlocality, and fractional operators are some of the important variables that contribute to its success, and this article provides a full study and illustration of the significance of these characteristics. The results that were obtained have the potential to play a significant role in pushing forward the design and development of tools, materials, and nanostructures that have viscoelastic mechanical characteristics and graded functions.

Vibration and Small Scale Effects of Skew Graphene Sheets Using Nonlocal Elasticity Theory

Nonlocal elasticity theory is a popularly growing technique for the realistic analysis of nano structures. In the present work nonlocal elasticity plate theory has been employed and vibration analyses of skew graphene sheets are carried out. Relevant governing differential equations are reformulated using the nonlocal differential constitutive relations suggested by Eringen. The equations of motion including the nonlocal theory are derived. All edges of the skew graphene sheets are assumed to be simply supported. Naviers approach has been employed to solve the governing differential equations. Bauers skew plate analysis has been extended to include the nonlocal elasticity plate theory. Vibration response of the skew graphene sheets is studied. Effects of the (i) size of the graphene sheets (ii) modes of vibration (iii) nonlocal parameter and (iv) skew angle of graphene sheet on nonlocal vibration frequencies are investigated. It has been observed that the vibration response of the skew graphene sheets are influenced significantly by the nonlocal parameter.

Small Scale Effects and Vibration of Graphene Sheets with Various Boundry Conditions

Elastic theory of graphene sheets is reformulated using the nonlocal differential constitutive relations of Eringen. The equations of motion of the nonlocal theories are derived. Levy’s approach has been employed to solve the governing differential equations for various boundary conditions. Nonlocal theories are employed to bring out the small scale effect of the nonlocal parameter on the natural frequencies of the graphene sheets. Present vibration results associated with various boundary conditions are in good agreement with those available in literature. Further, effects of (i) nonlocal parameter, (ii) size of the graphene sheets and (iii) boundary conditions on nondimensional vibration frequencies are investigated. The theoretical development as well as numerical solutions presented here in should serve as reference for nonlocal theories of nanoplates and nanoshells.

Buckling Analysis of Orthotropic Graphene Sheets Supported on Pasternak Elastic Medium Using Nonlocal Elasticity Theory and Differential Quadrature Method

In this paper, the small scale effect on the buckling analysis of bi-axially compressed orthotropic Single-Layered Graphene Sheets (SLGS) supported on elastic medium is studied. Elastic theory of the graphene sheets is reformulated using the nonlocal differential constitutive relations of Eringen. Both Winkler-type and Pasternak-type foundation models are employed to simulate the interaction between the graphene sheet and supporting elastic medium. Using the principle of virtual work the governing differential equations are derived for rectangular orthotropic graphene sheets supported on elastic medium. Solutions for buckling loads for various boundary conditions are computed using Differential Quadrature Method (DQM). Parametric study has been performed to investigate the dependence of small scale effect on various graphene sheet parameters. It is observed that the nonlocal effect is significant in graphene sheets supported on elastic medium and has a decreasing effect on the buckling loads.

Vibration analysis of piezoelectric semiconductor beams with size-dependent damping characteristic

In this paper, the vibration analysis of the piezoelectric semiconductor (PS) beam with size-dependent damping characteristic is conducted based on the nonlocal PS model. The nonlocal constitutive relations including the stress, electric displacement as well as current density of electrons are achieved. The one-dimensional vibration governing equations are obtained by integrating the three-dimensional governing equations along the cross section. Then, the natural frequency and damping characteristic are given by solving the governing equations of the PS beam with simply-supported boundary conditions. Through the numerical results, the effect of the nonlocal parameter, initial electron concentration, length to radius ratio and different scales on the vibration behavior of the system is demonstrated. The main innovation of the manuscript is that the peak value of the damping shifts to the smaller initial electron concentration because of the nonlocal effect, and the damping characteristics of the PS beam at nanoscale, micrometer scale and millimeter scale are obviously different.

Nonlinear dynamic modeling of geometrically imperfect magneto-electro-elastic nanobeam made of functionally graded material

Smart magneto-electro-elastic (MEE) structures with high performance are highly valuable across a vast range of aerospace and industrial applications like smart sensing, space robotics, energy harvesting and biomedical research. This work focuses on the nonlinear dynamic response of a functionally graded MEE nanobeam. Initial geometric imperfection including global and localized imperfection modes is considered to reflect the effect of undesired error during nanofabrication process. Such geometric imperfection is involved by the product of hyperbolic and trigonometric functions. The nonlinear dynamic model for the FG-MEE nanobeam is established according to Hamilton’s principle and nonlocal constitutive relation to derive its governing equations of motion. The interaction between the imperfect FG nanobeam and its multi-physical stimulus is numerically solved via differential quadrature method. After validating the proposed model with existing works, the parametric study is conducted to fully reveal the coupling effect of geometric imperfection and numerous parameters including imperfection mode and amplitude, external electric and magnetic potentials, functionally graded index, boundary constraint and size-dependent parameter on nonlinear dynamic response of the FG-MEE nanobeam. Simulation results have demonstrated the significance of the MEE coupling effect, which can be used as a guide to design smart and advanced devices for aerospace and industrial applications.

Non-local Effects in Inhomogeneous Flows of Soft Athermal Particles

We study non-local effects on inhomogeneous flows of soft athermal particles near the jamming transition. We employ molecular dynamics simulations to demonstrate Kolmogorov flows, where a sinusoidal flow profile with fixed wave number is externally imposed, resulting in a spatially inhomogeneous shear rate. We find that the rheology of soft athermal particles is strongly wave number-dependent and particle migration is not sufficient to explain the resulting stress profiles within conventional local constitutive relations. We show that stress profiles can be described by non-local constitutive relations that account for gradients to fourth order.

A physically-based nonlocal strain gradient theory for crosslinked polymers

When the length scale of external stimulations is comparable to most chain lengths within a polymeric solid, the nonlocal and microstructure-dependent strain-gradient effects become significant. The present work proposes a physically-based nonlocal strain gradient theory of polymer networks, where the kernel functions and intrinsic length scales have unambiguous physical meanings. The main contribution lies in the establishment of the general framework that takes in an arbitrary complete set of microscopic descriptions (chain energetics, chain-length distribution, structure of interpenetrating network) and outputs a corresponding nonlocal strain gradient constitutive relation. Based on the hypotheses of interpolatory correlation, homogenization, and superposition, the physically-based construction of constitutive relation is specified to a degree to uniquely determine two linear functionals, which are used to evaluate the nonlocal stress and nonlocal hyper-stress. Application to eight-chain interpenetrating networks yields a specific theory of nonlocal strain gradient elasticity in which the explicit functional form of the kernel is analytically derived from the chain length distribution function. It is shown that the physically derived kernel for the strain gradient field results from the superposition of a spectrum of length scales. The proposed physically-based nonlocal strain gradient theory can be beneficial to clarify the relationship between the microstructure-dependent intrinsic lengths and properties (or responses) of crosslinked polymer network structures.

Investigating parametric homogenization models for natural frequency of FGM nano beams

<abstract> <p>This research focuses on exploring the free vibration behavior of functionally graded (FG) nano-beams. To calculate the effective properties of the FG nano-beam, which varies solely in the thickness direction, the four homogenization schemes Mori-Tanaka, Tamura, Reuss and Voigt are employed. This study employs high-order shear deformation nano-beam theory and derives the governing equations of motion using nonlocal differential constitutive relations of Eringen. Hamilton's principle is utilized in conjunction with the refined three variables beam theory. The consideration of a length scale parameter accounts for small-scale effects. Analytical solutions are obtained for a simply supported FG nano-beam and compared with existing literature solutions. The research also investigates the influence of different homogenization schemes, the nonlocal parameter, beam aspect ratio and various material compositions on the dynamic response of the FG nano-beam.</p> </abstract>

On the Static Flexural Response of Shear Deformable Isotropic Rectangular Nanobeams under the Parabolic Loading

It is well known that size-dependent effects, which are insignificant for macroscopic structures, become significant for small-scale structures like nanobeams. In addition, effects of the beam transverse shear deformation, which are insignificant for slender beams, become significant for shear deformable beams. Hence the need arises for a beam theory which is simple-to-use as well as which accounts for just-mentioned features of shear deformable nanobeams. Recently, the authors have developed a single variable new first-order shear deformation nonlocal beam theory (NFSDNBT, doi: 10.1007/s40430-019-2128-6). The NFSDNBT is applicable for the flexure of linear isotropic nanobeams undergoing small deformations. Displacement functions of the NFSDNBT give rise to the constant transverse shear strain through the beam thickness. Hence similar to the nonlocal Timoshenko beam theory (NTBT), the NFSDNBT also requires a shear correction factor. In the NFSDNBT, nonlocal differential stress-strain constitutive relations of Eringen are utilized, through which size-dependent effects have been taken into account. These relations relate not only the beam axial stress with the beam axial strain but also the beam transverse shear stress with the beam transverse shear strain. The governing equation of the NFSDNBT is obtained by utilizing beam gross equilibrium equations and it has a strong resemblance with the governing equation of the nonlocal Bernoulli-Euler beam theory (NBEBT). In this paper, the NFSDNBT is utilized for finding the static flexural response of shear deformable isotropic rectangular nanobeams under the action of parabolically distributed transverse loading. For the simply-supported, clamped-clamped and cantilever nanobeams, effects of variations in values of the nonlocal parameter of Eringen and beam thickness-to-length ratio on the maximum non-dimensional beam transverse displacement are presented. Obtained results are compared with corresponding results obtained by utilizing the NTBT and NBEBT so as to demonstrate the efficacy of the NFSDNBT. Along-the-length profiles of the non-dimensional beam transverse displacement for the just-mentioned cases of nanobeams, for various values of the nonlocal parameter and beam thickness-to-length ratio are also presented.

Non-local multi-fields coupling response of a piezoelectric semiconductor nanofiber under shear force

Based on the nonlocal piezoelectric semiconductor theory, the transient multi-fields coupling response of a nanosized piezoelectric semiconductor composite fiber is studied. The one-dimensional phenomenological theory of piezoelectric semiconductors is used to derive the coupling constitutive relation. Combining nonlocal piezoelectric semiconductor theory and coupling constitutive relation, the size-dependent governing equations for bending, the charge equation of electrostatics and the conservation of charge for electrons are obtained, and the multi-fields coupling response of the nanofiber under shear force is solved. The bending vibration, the electric potential, and the concentration of electrons along the nanofiber with different nonlocal effects are analyzed. It is found that the change of electric potential and concentration from positive and negative in the nanofiber can be used to control the performance of electronic devices.

Dislocation singularities in layered magneto-electro-elastic plates

A general and unified theory is formulated to investigate static and time-harmonic field solutions induced by dislocation loops and dislocation arrays in three-dimensional multilayered structures. Each homogeneous plate consists of an orthotropic magneto-electro-elastic material including nonlocal effects. While the nonlocal constitutive relations with multi-phase coupling are treated by means of the original Eringen model using a Helmholtz-type operator, the field expressions are based on the mathematically elegant and computationally powerful Stroh formalism in matrix form, consistently combined with double Fourier series expansions and the dual variable and position technique to propagate the extended solutions among the different layers of the multilayered systems. The time-harmonic dislocation loops are represented by a discontinuity in the prescribed elastic displacement, electric potential, and magnetic potential on arbitrarily-located rectangular and elliptical surfaces in the multilayered structures, while the dislocation arrays are composed of infinitely long, straight and uniformly spaced parallel dislocations with the same local Burgers vectors. The new field solutions are first validated against existing frameworks limited to static and local elasticity theory of these two types of extrinsic and intrinsic dislocations, and subsequently applied to analyze several unexplored effects on the dislocation-induced magneto-electro-elastic fields, namely the material anisotropy, interaction with internal heterophase interfaces, multi-phase coupling, nonlocal core-spreading parameter, finite-valued driving forces, vibration frequency, and stacking sequences. The numerical outcomes indicate that each effect is significant and neglecting any one of them leads to an erroneous prediction on the extrinsic and intrinsic dislocation-induced response, thus providing a suitable route for the design of advanced magneto-electro-elastic fabrication devices for energy harvesting applications.

Foundations of volume integral methods for eddy current problems

Integral methods for solving eddy current problems use Biot–Savart law to produce non-local constitutive relations that lead to fully populated generalized mass matrices, better known as inductance matrices. These formulations are appealing because—unlike standard Finite Element solutions—they avoid the generation of a mesh in the insulating regions.The aim of this paper is to alleviate the three main problems of volume integral methods. First, the computation of the inductance matrix elements is slow and also delicate because of the singularity in the integral equation. This paper introduces novel face basis functions that allow a much faster inductance matrix construction with respect to the standard one based on the Rao–Wilton–Glisson (RWG) or Raviart–Thomas (RT) basis functions. Second, our basis functions work for polyhedral elements formed by any number of faces (including prisms, hexahedra and pyramids), while producing the same results as RWG and RT basis functions for tetrahedral meshes.Third, the new basis functions allow to factorize the inductance matrix and to introduce a novel family of groundbreaking low-rank inductance matrix compression techniques that show several orders of magnitude improvement in memory occupation and computational effort than state-of-the-art alternatives, allowing to solve problems that otherwise cannot be faced.

Nonlocal and magneto effects on dispersion characteristics of Love-type waves in piezomagnetic media

Attributions of intrinsic nanoscale dimension in piezomagnetic structures, the small scale effects on the propagating wave are undeniable in their piezotronic applications. In this work, based on the nonlocal constitutive relations, the nanoscale size effects on the Love-type wave propagating in piezomagnetic structure are investigated. Dispersion equations are derived for magnetically open and short circuit cases. Particular results are deduced and validated with the existing results. Variation of phase velocity and group velocity versus the penetration depth has been distinctly marked. Influence of various parameters like nonlocal parameter, magneto parameters, magnetomechanical coupling coefficient associated with piezomagnetic media etc. on the dispersion characteristics for first three modes has been demonstrated graphically. This research shows that small scale effects have substantial effect on the phase velocity of the propagating wave. The results find its enormous possible applications in smart devices especially, nanoscale devices using wave propagation phenomena owing to representing outstanding magnetic and mechanical coupling performances.

On size-dependent wave propagation of flexoelectric nanoshells interacted with internal moving fluid flow

In this study, dynamic properties of fluid-conveying flexoelectric cylindrical nanoshells subjected to an electric field are examined throughout the wave propagation approach. Employing nonlocal strain gradient theory (NSGT) and constitutive flexoelectric relations for shell-type structure, a new model has been represented regarding the Love shell theory. Governing motion equations are derived exploiting Hamilton’s principle and are then solved numerically to acquire the wave propagation properties. A discussion on numerical results is then implemented to highlight the exact wave propagation response of the structure against the flexoelectric effect, electric field, flow velocity and external force. The results reveal a remarkable effect of flexoelectricity on wave propagation response especially for some modes in specific wave number domains. As a prominent result, the flexoelectricity effect on wave dispersion response is found more pronounced for second and third modes, compared to the first mode. In addition, this effect is more tangible for lower wave numbers. The applied positive or negative electric and mechanical force exhibits a decreasing or increasing effect on the phase velocity of all the modes. The presented investigation and the reported results may be useful in designing the systems which are benefiting from wave propagation phenomenon properties of cylindrical nanoshells.

Analysis of the Vibration Behaviors of Rotating Composite Nano-Annular Plates Based on Nonlocal Theory and Different Plate Theories

Rotating machinery has significant applications in the fields of micro and nano meters, such as nano-turbines, nano-motors, and biomolecular motors, etc. This paper takes rotating nano-annular plates as the research object to analyze their free vibration behaviors. Firstly, based on Kirchhoff plate theory, Mindlin plate theory, and Reddy plate theory, combined with nonlocal constitutive relations, the differential motion equations of rotating functionally graded nano-annular plates in a thermal environment are derived. Subsequently, the numerical method is used to discretize and solve the motion equations. The effects of nonlocal parameter, temperature change, inner and outer radius ratio, and rotational velocity on the vibration frequencies of the nano-annular plates are analyzed through numerical examples. Finally, the relationship between the fundamental frequencies and the thickness-to-radius ratio of the nano-annular plates of clamped inner and outer rings is discussed, and the differences in the calculation results among the three plate theories are compared.

Analytical approach to determine the impact of line source on SH-wave propagation in an anisotropic poro-viscoelastic layered structure in the context of Eringen's nonlocal elasticity theory

The present contribution investigates the influence of line source on the propagation of Horizontally polarized shear waves (SH-waves) in a complex stratified structure which comprises two distinct nonlocal anisotropic fluid-saturated poro-viscoelastic layers resting over a nonlocal functionally graded anisotropic fluid-saturated poro-viscoelastic substrate. Based on Eringen's nonlocal elasticity theory, constitutive relations and equations of motion for a nonlocal anisotropic poro-viscoelastic medium are developed in this article. Green's function technique and Fourier transformation are applied in this study to acquire the complex frequency relation of the propagating waves in the context of suitable boundary conditions. Dissociation of the complex frequency equation into real and imaginary parts leads to two distinct equations featuring dispersion and attenuation properties of SH-waves in the considered model, respectively, which represent significant relevance to the field of geophysics, earthquake engineering and civil engineering. As a special case of the problem, the closed form of the dispersion equation has been derived for a structure with a single porous layer overlying a semi-infinite medium which has been validated with the pre-established result and the final particular case is in well agreement with the classical Love wave equation. Several graphs are plotted to demonstrate the impact of functionally graded parameter, ratio of widths of two layers, quality factors, and dissipation parameter on the dispersion and attenuation of SH-waves. Moreover, the graphical implementation of the variation in shear wave speeds of each medium due to the change in nonlocal parameter is one of the major highlights of this study.

Higher order constitutive relations and interface conditions for metamaterials with strong spatial dispersion

To characterize electromagnetic metamaterials at the level of an effective medium, nonlocal constitutive relations are required. In the most general sense, this is feasible using a response function that is convolved with the electric field to express the electric displacement field. Even though this is a neat concept, it bears little practical use. Therefore, frequently the response function is approximated using a polynomial function. While in the past explicit constitutive relations were derived that considered only some lowest order terms, we develop here a general framework that considers an arbitrary high number of terms. It constitutes, therefore, an approximation to the initially considered response function of arbitrary precision. The reason for the previously self-imposed restriction to only a few lowest order terms in the expansion has been the unavailability of the necessary interface conditions with which these nonlocal constitutive relations have to be equipped. Otherwise one could not make practical use of them. Therefore, besides the introduction of such higher order nonlocal constitutive relations, it is at the heart of contribution to derive the necessary interface conditions to pave the way for the practical use of these advanced material laws.

On well-posedness of two-phase nonlocal integral models for higher-order refined shear deformation beams

Due to the conflict between equilibrium and constitutive requirements, Eringen’s strain-driven nonlocal integral model is not applicable to nanostructures of engineering interest. As an alternative, the stress-driven model has been recently developed. In this paper, for higher-order shear deformation beams, the ill-posed issue (i.e., excessive mandatory boundary conditions (BCs) cannot be met simultaneously) exists not only in strain-driven nonlocal models but also in stress-driven ones. The well-posedness of both the strain- and stress-driven two-phase nonlocal (TPN-StrainD and TPN-StressD) models is pertinently evidenced by formulating the static bending of curved beams made of functionally graded (FG) materials. The two-phase nonlocal integral constitutive relation is equivalent to a differential law equipped with two restriction conditions. By using the generalized differential quadrature method (GDQM), the coupling governing equations are solved numerically. The results show that the two-phase models can predict consistent scale-effects under different supported and loading conditions.

Lower limits for the homogenization of periodic metamaterials made from electric dipolar scatterers

Nonlocal constitutive relations promise to homogenize metamaterials even though the ratio of period over operational wavelength is not much smaller than unity. However, this ability has not yet been verified, as frequently only discrete structures were considered. This denies a systematic variation of the relevant ratio. Here, we explore, using the example of an electric dipolar lattice, the superiority of the nonlocal over local constitutive relation to homogenize metamaterials when the period tends to be comparable to the wavelength. Moreover, we observe a breakdown of the ability to homogenize the metamaterial at shorter lattice constants. This surprising failure occurs when energy is transported across the lattice thanks to a well-pronounced near-field interaction among the particles forming the lattice. Contrary to common wisdom, this suggests that the period should not just be much smaller than the operational wavelength to homogenize a metamaterial, but, for a given size of the inclusion, there is an optimal period.