Nonlocal Parameter Research Articles

The use of a nonlocal critical state model in modelling triaxial and plane strain tests on overconsolidated clays

When modelling the phenomenon of strain localisation in strain-softening soils with the finite element (FE) method, nonlocal approaches have been commonly employed to avoid mesh dependency and numerical instability. This paper first presents the FE formulation of a critical state model for highly overconsolidated clays incorporating a nonlocal method. The performance of the nonlocal strain regularisation is subsequently assessed through a series of coupled hydro-mechanical (HM) analyses of undrained and drained triaxial compression tests on London clay. The mechanism behind the evolution of strain localisation in triaxial tests is investigated and a comparison with equivalent plane strain analyses is discussed. Finally, a comprehensive sensitivity study is presented, investigating the influence of the two nonlocal parameters, in the adopted nonlocal algorithm, on the predicted stress–strain responses. A key outcome is the derived linear relationship between the two parameters, which enables a unique stress–strain response to be achieved in either axisymmetric or plane strain analyses with multiple combinations of the two parameters. Such a modelling capability is essential in applications of the proposed nonlocal strain regularisation in large scale boundary value problems in which restrictions on element size exist.

Electro-mechanical vibration characteristics of imperfect sandwich FG piezoelectric honeycomb nanoplates resting on Kerr foundation

This study presents an innovative analytical approach to examine the electro-mechanical vibration characteristics of sandwich nanoplates featuring a honeycomb core and two surface layers composed of imperfect piezoelectric functionally graded (FG) material, incorporating porosities, atop a Kerr elastic foundation. Utilizing nonlocal elasticity theory and four-variable refined plate theory (HSDT-4), we account for nanoscale effects and model the electric potential variation across the piezoelectric layer thickness as a combination of cosine and linear variations satisfying Maxwell's equation. The proposed method's novelty lies in its comprehensive analysis of the interplay between porosity, elastic foundation parameters, material properties, geometrical dimensions, external voltage, and nonlocal parameters on the free vibration frequency of these nanostructures. Numerical simulations validate the model's reliability and assess the significant impacts of these factors. Our findings contribute to the optimization of nanostructural designs for advanced engineering applications, highlighting the method's added value in predicting the behavior of imperfect piezoelectric FG honeycomb nanoplates.

Generalized thermoelastic damping in micro/nano-ring resonators undergoing out-of-plane vibration

Micro/nano-ring or ring-based resonators are popular core components of sensors and actuators. Accurately predicting the quality factor (Q) based on thermoelastic damping (TED) is intensively crucial for designing high-performance micro/nano-ring resonators. In this work, adopting the modified-couple-stress (MCS) and nonlocal-dual-phase-lagging (NDPL) models, analytical TED expressions are developed for circular cross-section micro/nano-rings undergoing out-of-plane vibration for the first time. Multiple effects are considered in the modeling procedure, which are size-dependence effects in the mechanical and thermal fields, the non-Fourier effect of heat conduction, and three-dimensional temperature gradients in the circumferential, radial, and tangential directions. Therefore, the proposed TED models are more comprehensive compared to existing TED models. The coefficients (CC and CM) associated with the out-of-plane vibration mode are examined from an energy-ratio perspective. Moreover, the investigation is focused on the impacts of the MCS and NDPL non-Fourier effects on the fluctuation temperature and TED spectra. The results indicate as CC and CM approach one, the bending energy emerges as the predominant component of the stored energy. Notably, the NDPL non-Fourier effect significantly affects the fluctuation temperature within the ring operated in high-order modes. In special, the distributions of fluctuation temperature in the circumferential direction exhibit periodic patterns that are closely correlated with the modal order. Furthermore, TED can be suppressed by the MCS size-dependence effect. Meanwhile, the behavior of TED at high frequencies is simultaneously determined by the dimension of heat conduction, phase-lagging times, and nonlocal thermal-length parameter.

Nonlocal strain gradient-based geometrically nonlinear vibration analysis of double curved shallow nanoshell containing functionally graded layers

It is a fact that while examining the mechanical behavior of nano/microscopic materials, size effects become apparent. This study aims at presenting a new approach to geometrically nonlinear vibration analysis of double curved shallow nanoshell on Kerr foundation containing functionally graded layers under the impacts of harmonic loading in the framework of nonlocal strain gradient theory and the classical shell theory. The three-parameter Kerr model is founded on the independence of the upper and lower elastic layers related to the shear layer. The motion equations of the double curved shallow nanoshell are deduced by introducing the Von Kármán strain-displacement relations. Governing equations are solved by displacement function method and Galerkin method to achieve the deflection differential equation, the fourth-order Runge-Kutta methods are implemented to solve the deflection differential equation of the dynamic system. In this study, the obtained outcomes are compared with other publications indicating that the theoretical results agree with the previously published ones. Afterward, the article investigates the effects of the material length scale parameter, the nonlocal parameter, the geometric parameters and the material properties on the geometrically nonlinear vibration analysis of the double curved shallow nanoshell containing functionally graded layers. This research paper can provide extensive references and valuable guidelines for the impact and application of double curved shallow nanoshell structures containing functionally graded layers.

An analytical model for Love wave in a coated piezoelectric bar via nonlocal theory due to an impulsive source

The non-locality behavior of new advanced functionally graded (FG) and piezoelectric (PE) nanobar is presented in this study with an impulsive point source using green function derivatives. The primary aim of this study is to explore the impact of different physical parameters on the phase dispersion phenomena against wave number and depth of point source of Love wave propagation. The stress tensor is considered the linear variation of the shear stress distribution in the thickness direction, and the shear stress at the free surface of the piezoelectric bar is zero. Using the Fourier transform and Green’s function technique, the equations of motion and the dispersion relation is derived by applying admissible boundary conditions involving piezoelectricity, dielectricity, functional gradient, and nonlocal elastic parameters of the media. The obtained results are validated with existing results by considering special cases. The variation in the dispersion curves due to the impulsive point source and piezoelectricity, dielectricity, functional gradient, and non-locality parameters have been analyzed and illustrated using graphs along with a comparative study by considering four piezoelectric materials, PZT-4, PZT-5A, PZT-5H, and PZT-2. Furthermore, the relationship between nonlocal parameters in the layer and half-space is investigated to explore the influence of wave number, phase velocity, and depth of point source.

Thermoelastic diffusion based on a nonlocal three-phase-lag diffusion model with double porosity structure

This work aims to develop a new diffusion-elasticity-based model using Eringen’s nonlocal elasticity theory under the purview of the three-phase-lag (TPL) model of hyperbolic thermoelasticity. This problem is treated in the context of a double porosity structure in a homogeneous, isotropic thermoelastic medium. By considering the phase laggings of the diffusion flux vector and using nonlocal continuum mechanics, new constitutive and field equations are derived. The problem is solved by employing the normal mode analysis technique which gives the exact results of all the physical variables. To illustrate the theoretical results, different physical quantities are calculated numerically and presented graphically with respect to distance and time. The influences of different thermoelastic models like TPL (three-phase-lag model), GN-III (Green-naghdi model), DPL (dual-phase-lag model), LS (Lord-Shulman model), and CT (coupled thermoelasticity) on the physical quantities are shown graphically. Additionally, the effects of nonlocal parameters, double voids, and diffusion on the physical quantities are represented graphically. Some limiting and particular cases are also derived from the governing equations and those results have been compared with the existing results available in the literature. Some 3D surface curves are also presented to study the impact of diffusion, nonlocality, and double voids on various physical quantities.

The role of nonlocality on low and high frequency behaviors of functionally graded sandwich nanoplates

AbstractThe role of nonlocality on low and high frequency behaviors and modes of the functionally graded (FG) sandwich nanoplates is investigated in this study for the first time using simple nonlocal higher‐order shear deformation theory (HSDT). The simple HSDT consists of only four unknowns in its equation of the displacement field. The nonlocal elasticity theory is used to consider the small‐scale effects on the behaviors of the nanoplates. The governing equations of motion are established by applying Hamilton's principle, then Navier's solution technique is employed to solve these equations to achieve the free vibration behaviors of the FG sandwich nanoplates. The vibrations of the nanoplates under both low and high frequency conditions are investigated, but the high frequency vibration of the nanoplates is discussed extensively. The influences of the geometrical dimensions, material gradient index, and nonlocal parameter on the high frequency vibration of the FG nanoplates are also considered and discussed comprehensively.

Wave propagation in context of Moore–Gibson–Thompson thermoelasticity with Klein–Gordon nonlocality

Understanding plane and surface waves in elastic materials is crucial in various fields, including geophysics, seismology, and materials science, as they provide valuable information about the properties of the materials they travel through and can help in earthquake detection and analysis. In the present paper, the governing equations of Moore–Gibson–Thompson (MGT) thermoelasticity are modified in context of Klein–Gordon (KG) nonlocality. For linear, homogeneous and isotropic case, the governing equations in two-dimensions are solved to obtain the dispersion relations for possible plane waves. It is found that there exists one transverse and two coupled longitudinal waves in a two-dimensional model of MGT weakly nonlocal thermoelastic medium and the speeds of these plane waves are found to be dependent on KG nonlocal parameters. The coupled longitudinal waves are also found to be dependent on conductivity rate parameter. For linear, homogeneous and isotropic case, the governing equations in two-dimensions are also solved to obtain a Rayleigh wave secular equation at thermally insulated surface. For a numerical example of aluminium material, the speeds of transverse wave, coupled longitudinal waves and the Rayleigh wave are computed and graphically illustrated to visualize the effects of KG nonlocality parameters, conductivity rate parameter and the angular frequency on the wave speeds.

Dynamic Analysis of Functionally Graded Sandwich Doubly Curved Nanoshells Using Variable Nonlocal Elasticity Theory

Sandwich nanoshells consisting of functionally graded materials (FGMs) offer promise in aerospace, energy and biomedical applications due to their tunable properties. However, a comprehensive understanding of the vibration behavior of such shells is lacking. This work develops an efficient analytical model to study free vibration of sandwich nanoshells with three FGM configurations: FGM face-sheets and metal core (Type A), FGM face-sheets and ceramic core (Type B), and FGM core and face-sheets (Type C). A doubly curved shell geometry representing spherical, hyperbolic–parabolic and cylindrical shells in addition to flat plates is considered. Effective material properties of nanolaminates are computed using a volume fraction-based model. First-order shear deformation theory and modified nonlocal elasticity theory with variable nonlocal parameters are employed to derive the governing equations. Hamilton’s principle is used to obtain the equations of motion. Navier’s solution technique analytically solves the equations for simply supported boundary conditions. Accuracy is validated through comparisons with literature results. Parametric studies analyze the influence of thickness ratio, material gradation, nonlocal parameter, and shell type on frequencies. Results show that Type A configuration yields the highest frequencies. Dimensionless frequency decreases with increasing nonlocal parameter and side-to-thickness ratio. The model provides design insights into vibration behavior of such nanosandwich structures with potential applications in nanodevices and energy harvesting.

Free Vibration of Thermally Loaded FG-GPLRC Nanoplates Integrated with Magneto-Electro-Elastic Layers in Contact with Fluid

This work investigates the free vibrations of innovative thermally loaded nanoplates constructed by integrating magneto-electro-elastic (MEE) layers with functionally-graded graphene platelet-reinforced composite cores (FG-GPLRC) and accounting for viscous fluid interactions. An advanced multiphysics model is developed using the Navier–Stokes equations to capture fluid structure coupling effects, Halpin–Tsai, and the rule of mixtures micromechanics to predict the non-uniform effective properties, third-order shear deformation plates theory (TSDPT) to incorporate thickness stretching, and the nonlocal strain gradient theory (NSGT) to characterize size dependencies. The Galerkin technique is used to solve the governing equations, which are derived from the Hamilton’s principle. Parametric analyses quantify the influences of fluid depth, temperature fluctuations, temperature profiles, nonlocal and strain gradient parameters, electric and magnetic potentials, graphene distribution patterns, graphene weight fractions, and boundary conditions on the vibration response. The outcomes of this study provide design guidelines and predictive tools enabling active vibration control systems for next-generation thermally-loaded nanocomposite structures with widespread applications from aerospace vehicles to nanoelectronics.

Vibration of embedded restrained composite tube shafts with nonlocal and strain gradient effects

Torsional vibration response of a circular nanoshaft, which is restrained by the means of elastic springs at both ends, is a matter of great concern in the field of nano-/micromechanics. Hence, the complexities arising from the deformable boundary conditions present a formidable obstacle to the attainment of closed-form solutions. In this study, a general method is presented to calculate the torsional vibration frequencies of functionally graded porous tube nanoshafts under both deformable and rigid boundary conditions. Classical continuum theory, upgraded with nonlocal strain gradient elasticity theory, is employed to reformulate the partial differential equation of the nanoshaft. First, torsional vibration equation based on the nonlocal strain gradient theory is derived for functionally graded porous nanoshaft embedded in an elastic media via Hamilton’s principle. The ordinary differential equation is found by discretizing the partial differential equation with the separation of variables method. Then, Fourier sine series is used as the rotation function. The necessary Stokes' transformation is applied to establish the general eigenvalue problem including the different parameters. For the first time in the literature, a solution that can analyze the torsional vibration frequencies of functionally graded porous tube shafts embedded in an elastic media under general (elastic and rigid) boundary conditions on the basis of nonlocal strain gradient theory is presented in this study. The results obtained show that while the increase in the material length scale parameter, elastic media and spring stiffnesses increase the frequencies of nanoshafts, the increase in the nonlocal parameter and functionally grading index values decreases the frequencies of nanoshafts. The detailed effects of these parameters are discussed in the article.

Love-like wave fields at the interface of sliding contact with non-local elastic heterogeneous fluid-saturated fractured poro-viscoelastic layer

The investigation delves into the propagation of Love-like waves within a stratified medium incorporating the impact of non-local elasticity and a sliding interface. The medium is characterized by an orthotropic viscoelastic layer with fractures and matrix porosity saturated with fluid positioned over an orthotropic viscoelastic half-space under initial stresses. The media is vertically heterogeneous with a binomial function of depth featuring a real positive exponent. Two distinct non-local parameters are considered for the layer and half-space, along with a sliding parameter accounting for interface sliding. The particle displacement components are expressed using modified Bessel functions of first and second kinds. A finite number of terms of the asymptotic representation of the modified Bessel functions and their derivatives have been used and compared with the exact solution. The dispersion equation is derived by neglecting fourth and subsequent powers of non-local parameters under suitable boundary conditions. Various cases are explored based on welded, partially, and fully sliding interfaces and compared with classical Love wave propagation. An interesting phenomenon has been found that under a fully sliding interface, the existence of Love-like wave vanishes, and the medium behaves as a separate layer and half-space with shear wave propagation.

Propagation of plane waves at the initially stressed surface of an orthotropic nonlocal rotating half space under dual-phase-lag model

PurposeThe purpose of the current manuscript is to investigate the reflection of plane waves in a rotating, two-dimensional homogeneous, initially stressed, nonlocal orthotropic thermoelastic solid half-space based on dual-phase-lag model.Design/methodology/approachThe reflection phenomenon has been utilized to study the effects of initial stress, rotation and nonlocal parameter on the amplitude ratios. During the reflection phenomenon three coupled waves, namely quasi displacement primary wave (qP), quasi thermal wave (qT) and quasi displacement secondary wave (qSV) have been observed in the medium, propagating with distinct velocities. After imposing the suitable boundary conditions, amplitude and energy ratios of the reflected waves are obtained in explicit form.FindingsWith the support of MATLAB programming, the amplitude ratios and energy ratios are plotted graphically to display the effects of rotation, initial stress and nonlocal parameters. Moreover, the impact of anisotropy and phase lags is also observed on the reflection coefficients of the propagating waves.Originality/valueIn the current work, we have considered rotation and nonlocality parameters in an initially stressed orthotropic thermoelastic half-space, which is lacking in the published literature in this field. The introduction of these parameters in a nonlocal orthotropic thermoelastic medium provides a more realistic model for these studies. The present work is valuable for the analysis of orthotropic thermoelastic problems involving rotation, initial stress and nonlocality parameters.

Effect of the porous structure on the hygrothermal vibration analysis of functional graded nanoplates using nonlocal high-order continuum plate model

This study delves into the thermomechanical vibration behavior of functionally graded porous nanoplates under extreme thermal temperature and humidity conditions. The equation of motion of the nanoplate was derived using advanced theories in elasticity and deformation. The nanoplate consists of metal (SUS304) on the bottom surface and ceramic (Ni3S4) on the top surface, with the material distribution changing according to the power law across the plate thickness. The nanoplate was modeled with uniform and symmetric distributions of porosity reaching as high as 60%. Upon incorporating the thermal and moisture loads from the humid surroundings into the equations of motion derived from Hamilton's principle, the equations were solved using Navier's method and simplified to the eigenvalue equation. Analyzed within a broad framework are the thermomechanical vibration behavior of the nanoplate, temperature impact, humidity influence, porosity and its distribution, material grading parameter effects, and nonlocal integral elasticity effects. Observations indicate that variations in thermal temperature, humidity, and nonlocal parameters can lower the thermomechanical vibration frequency of the nanoplate, whereas porosity has the opposite effect. The effects mentioned are influenced by factors, such as the porosity ratio, porosity distribution, material ratios, and the size of the nonlocal parameter in the plate. The primary objective of this work is to uncover the nonlinear frequency response of nanoplates with high porosity in conditions characterized by high temperature and humidity.

Nonlinear vibration of five-layered functionally graded piezoelectric semiconductor nano-plate on Pasternak foundation

The vibration character of piezoelectric semiconductor nano-plate is important for understanding the multi-fields coupling and optimizing the serving behavior. In this article, the nonlinear vibration of five-layered functionally graded piezoelectric semiconductor nano-plate based on Pasternak foundation is studied, and the nonlocal theory is used to analyze the size-dependent dynamic response of nano-plate. Combining the Von karman’s nonlinear deformation theory and the constitutive of piezoelectric semiconductor layers, Hamilton’s principle is introduced to derive the nonlinear governing equations. To solve the nonlinear governing equations, the updated iteration method for this problem is proposed. Through numerical examples, it is found that the initial electron concentration in piezoelectric semiconductor layer, the nonlocal parameter, the functionally graded index in the plate, the elastic modulus of Pasternak foundation and the geometrical parameters can be designed to effectively tune the dynamic nonlinear frequency and damping of layered functionally graded piezoelectric semiconductor nanoplate. The interaction of these parameters is also analyzed in detail. Comparison with existing numerical results is also presented. A new way is provided to tune the nonlinear vibration of multi-layered piezoelectric semiconductor nano-plate.

Investigation of accelerated moving load on dynamic response of FG Timoshenko nanobeam in thermal environment based on nonlocal strain gradient theory

For the first time, this paper investigates the forced vibrations of a functionally graded (FG) nanobeam with an accelerating moving load in a thermal environment. There is no exact coupling solution for the vibrations of nanobeam with an accelerated moving load, so this paper aims to provide a method to obtain an accurate solution for nanoscale structures with an accelerated moving load. The equations of motion are derived using Timoshenko's beam theory and the nonlocal strain gradient theory (NSGT). The Laplace method has been used to solve the coupling and exact differential equations. Then, by inverting Laplace from the coupled equations, an exact solution of the temporal response for FG nanobeam with constant acceleration and initial velocity in a thermal environment was obtained. The natural frequency was compared with previous works for validity and had acceptable results. Finally, the effect of parameters such as changes in acceleration and velocity of moving force, negative acceleration, power law index of FG material, different temperatures, nonlocal parameter, and longitudinal scale parameter on the maximum dynamic displacement of nanobeam is investigated. These results can be used better to design FG nanostructures with accelerated moving loads. Considering the sensitivity of data analysis in nano dimensions, it is necessary to provide an analytical solution method in these dimensions to reduce the error percentage in nano dimensions to zero. which is presented in this work for the first time.

Fundamental role of nonlocal orders in 1D extended Bose-Hubbard model.

Nonlocal order parameters capture the presence of correlated fluctuations between specific degrees of freedom, in otherwise disordered quantum matter. Here, we provide a further example of their fundamental role, deriving the ground state phase diagram of the filling one extended Bose-Hubbard model, exclusively in terms of their ordering. By means of a density matrix renormalization group numerical analysis, we show that in addition to the (even) parity order characteristic of the Mott insulating phase and the string order nonvanishing in the Haldane insulator, the recently proposed odd parity order completes the picture, becoming nonvanishing at the transition from the normal superfluid to the paired superfluid phase. The above three nonlocal parameters capture all the distinct phases, including the density wave phase, in which the local order is seen as the simultaneous presence of correlated fluctuations in different channels. They provide a unique tool for the experimental observation of the full phase diagram of strongly correlated quantum matter, by means of local density measurements.

On nonlinear buckling of microshells

Investigation of the geometrical nonlinear action of doubly curved shell panels (DCSPs) in micro scale is the main target of this paper. The proposed microshell panels (MSPs) are assumed to be made of an auxetic honeycomb core (AHOC), leading to negative magnitudes of Poisson's ratio, covered by two nanocomposite enriched coating layers (NCECLs). To conduct the size-dependent nonlinear analysis and achieve the corresponding nonlinear equilibrium path (EQP) of the proposed MSPs, the nonlocal strain gradient theory (NLSGT) is utilized. The governing equations containing the equilibrium and compatibility nonlinear partial differential equations in terms of the deformation components are analytically solved based on the Galerkin technique for different types of simply-supported panels. The achieved results of the present solution exhibit the fact that nonlocal and material length scale parameters significantly affect the EQP of the proposed MSPs especially at their post-buckling stage during their snap-through instability. By solving several numerical examples, the effects of various parameters on the size-dependent EQP of the proposed MSPs are investigated. The results indicate that the influences of size-dependency are significantly affected by the curvature and also boundary conditions of the microshells.

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.

Electromechanical coupling characteristics of multilayered piezoelectric quasicrystal plates in an elastic medium

AbstractQuasicrystals (QCs) have attracted tremendous attention of researchers for their unusual properties. In this paper, an exact electric‐elastic solution of the simply supported and multilayered three‐dimensional (3D) cubic piezoelectric quasicrystal (PQC) nanoplate with the nonlocal effect is derived. Based on the basic elasticity equation of 3D QCs, we construct the linear eigenvalue system in terms of the pseudo‐Stroh formalism, from which the general solutions of the extended displacements and stresses in any homogeneous layer can be obtained. The two‐parameter foundation model is utilized to simulate the interaction between the nanoplate and elastic medium. The propagator matrices are employed to connect the field variables at the upper interface to those at the lower interface of each layer. Based on the boundary conditions of the upper and lower surfaces of the laminate and foundation model, the solutions are employed to derive from the global propagator matrix. Compared with the conventional propagator matrix method, a new propagator method is reestablished to deal with numerical instabilities of the case of large aspect ratio and high‐order frequencies for QC laminates. Finally, typical numerical examples are presented to illustrate the influence of nonlocal parameters and elastic medium coefficients on phonon, phason, and electric variables of 3D PQC nanoplates.