Nonlocal Elasticity Theory Research Articles

Dynamic stiffness formulation for the vibration response of functionally graded nanoplates considering the neutral surface position with microstructural defects

In the present paper, the dynamic stiffness matrix of a functionally graded nanoplate (FG-nP) has been developed and used to carry out a vibration analysis. The nonlocal elasticity theory in conjunction with classical plate theory (CPT) has been used to drive the governing equations of motion using Hamilton’s principle. The nonlocal elasticity theory incorporates the length scale parameter, which can withhold the small-scale effect that is necessary for model FG-nP. The pioneer Wittrick–Williams algorithm is employed to evaluate the dynamic stiffness matrix. The property of the FG-nP has been calculated using the power law. The authenticity and efficacy of the present formulation have been ascertained using various validation studies. The accurate prediction of the effect of porosity inclusions on different volume fraction indexes, boundary conditions, and higher vibration modes has been reported. The influence of the nonlocal parameter, boundary conditions, various gradation techniques, and geometric parameters on vibration behavior has been extensively explored. The results show the versatility of the proposed approach in addressing small scaled complex engineering structures. This research significantly contributes to the understanding and analysis of functionally graded nanoplates, providing valuable insights for applications in aerospace, structural systems, sensors, actuators, and energy harvesting devices.

Dynamic analysis of viscoelastic functionally graded nanoplate

In this article, the dynamic behavior of nanoplates under time-dependent load is investigated, focusing on functionally graded viscoelastic materials and nanoscale effects. Eringen’s nonlocal elasticity theory is utilized to examine mechanical response of the nanoplate. Hamilton’s principle is utilized to derive the equations of motion, taking into account both kinetic and potential energy aspects. The obtained complex partial differential equations are then solved employing Navier method and provides an efficient way to obtain analytical solutions. The study initially performed a free vibration analysis for functionally graded nanoplate, comparing the obtained results with those available in the literature to validate the developed method. Following this validation, a parametric analysis was conducted to examine the influence of both nonlocal parameter, which accounts for nanoscale effects, and power law exponent governing material gradation on free vibration behavior of functionally graded nanoplate. Finally, as the original contribution of this study, a damped forced vibration analysis was carried out within the scope of the parametric study, investigating the effects of power law exponents, viscoelastic parameters, nonlocal parameters, and various geometric properties on functionally graded viscoelastic nanoplates’ the displacement-time relationship and maximum displacements.

Stability and bifurcation analysis of simply supported rectangular nanoplate embedded on visco-Pasternak foundation based on first-order shear deformation theory

Vibrations of simply supported nanoplates embedded on visco-Pasternak foundation and stressed by in-plane periodic forces are investigated. The governing size-dependent equations employ the first-order shear deformation theory, nonlinear von Kármán strains, and the nonlocal elasticity theory. Reducing the governing system is carried out by the Bubnov-Galerkin method, which is based on two-mode model, thus the resulting system contains two ordinary differential equations with size- and time-dependent coefficients. Analysing the linearized system, the stability of the nanoplate structure is studied depending on the foundation parameters, force parameters and small-scale parameter. Nonlinear dynamics approaches are employed to determine the nature of vibration after the loss of stability. Time history, Poincare section, Lyapunov exponent are analysed for chosen values of excitation parameters, visco-Pasternak foundation parameters to indicate small-scale effects and nonlinear effects. The novelty of the work consists of the combined application of the nonlocal theory, first-order shear deformation theory, Floquet theory, methods of nonlinear dynamics and obtaining new results by a numerical experiment.

Axial Vibration of a Viscoelastic FG Nanobeam with Arbitrary Boundary Conditions

ObjectiveThis study investigates the axial vibration of a viscoelastic functionally graded (FG) nanobeam under deformable boundary conditions for the first time. The primary focus is on exploring the effects of damping and scale parameters on the dynamic behavior of the nanobeam.MethodsThe governing equation of the viscoelastic FG nanobeam is formulated by incorporating nonlocal elasticity theory and the Kelvin-Voigt viscoelastic model. The Fourier sine series is chosen as the axial displacement function, with higher-order derivatives obtained using Stokes transforms. The Fourier coefficient is determined through the governing equation and incorporated into the deformable boundary conditions. The resulting eigenvalue problem provides solutions for both rigid and constrained general boundary conditions.ConclusionsThe study presents solutions for various boundary conditions, comparing the results with existing literature. The analysis reveals significant findings, including the observation that damping has a greater influence on fundamental frequencies in higher modes, and that the impact of damping decreases as the nonlocal scale parameter increases. These findings are presented through tables and graphs to highlight the effects of damping and scale parameters on the dynamic behavior of the nanobeam.

A new shear deformation micro-beam model based on a nonlocal elasticity theory

In this article, a nonlocal refined shear deformation beam theory is proposed to analyze the free vibration behavior of FG nanobeams. The formulation is based on the nonlocal differential constitutive equations of Eringen that accounts for small scale effects. Material properties vary continuously through thickness according to a power law distribution. The study is performed within the framework of a new parametric shear deformation theory which provides parabolic transverse shear strains across the thickness direction and hence, does not need shear correction factor. Moreover, zero-traction boundary conditions on the top and bottom surfaces of the nanobeam are satisfied rigorously. Equations of motion of FG nanobeam are derived from both the nonlocal differential constitutive relations of Eringen and Hamilton’s principle. The system of differential equations is solved using the Navier solution technique. The parametric study is conducted to examine the effects of volume fraction index, length scale parameter and length ratio on the free frequencies of vibration of FG thin and thick nanobeams.

Finite element analysis of double nanobeams having Pasternak foundation in between

Single/double nanobeams (nanowires) have attractive features including reduced sizes and high flexibility and conductivity. As a result, many engineering applications such nanoelectromechanical systems (NEMS) and biomedical devices have been developed in technology industries. Due to the extremely high surface area-to-volume ratio, the properties of nanobeams have size-dependent behavior. In this paper, elastically connected double nanobeams are represented on Eringen’s nonlocal elasticity theory. Each nanobeam of double beam is modeled as Euler-Bernoulli beam and the interconnecting layer is represented by a Pasternak’s elastic foundation model. A four-node double-beam finite element with eight degrees of freedom using approximate functions to interpolate transverse displacements and rotations is derived where both stiffness matrix and load vector are explicitly shown. The present FEM solution is validated by numerical examples where the influence of effects of dimensionless small-scale parameters, boundary conditions, and shear parameter of Pasternak foundation on displacements and stress resultants of double nanobeams are investigated.

Stress-driven nonlocal integral model with discontinuities for transverse vibration of multi-cracked non-uniform Timoshenko beams with general boundary conditions

We present a formulation for the size-affected vibration study of multi-cracked non-uniform Timoshenko beams based on the well-posed stress-driven nonlocal elastic theory with discontinuities. The beam ends are assumed to be constrained by elastic springs with translational and rotational stiffness to simulate general boundary conditions. The presence of cracks divides the beam into segments connected by translational and rotational springs, and compatibility conditions are established to address the geometric discontinuities introduced by these cracks. The stress-driven constitutive equations are integrated into an equivalent differential form, equipped with a set of constitutive boundary conditions at the two ends of the entire structure and multi-sets of constitutive continuity conditions at the junctions of the sub-structures. To solve the equations of motion, the constraint conditions and the integrals involved, we employ the differential quadrature method (DQM) alongside an interpolation quadrature formula, which allows us to efficiently compute the frequencies of the cracked beams across various boundary types. After validating our approach against results in the existing literature, we present numerical studies that examine the effects of the nonlocal parameter, the slope of the beam’s thickness variation, crack location, severity, number, and the stiffness of the springs on the vibrational behavior of the beams.

Buckling analysis of single‐walled carbon nanotubes using the initial value method and approximate transfer matrix based on nonlocal elasticity theory

AbstractIn this study, a novel method is presented to analyze the buckling behavior of single‐walled carbon nanotubes (SWCNTs) using the initial value method (IVM) in conjunction with the approximate transfer matrix, within the framework of nonlocal elasticity theory. The study aims to accurately approximate critical buckling load parameters under various boundary conditions, without encountering high computational requirements. IVM enables the computation of displacements and stress resultants along the entire beam from given initial conditions. The approximate transfer matrix is employed to analyze system states at different points through successive integration of solutions, generating the principal matrix needed for IVM and ensuring systematic and precise results that optimize the accuracy of the analysis. A convergence study confirms the effectiveness and precision of the proposed method, revealing a decrease in the critical buckling load parameters as the nonlocal parameter increases, applicable across all boundary conditions studied (simply supported, clamped–clamped, clamped–simply supported, and clamped‐free). These results underscore the need to incorporate nonlocal effects for more accurate nanostructure mechanics predictions. The integration of IVM and the approximate transfer matrix provides a computationally efficient alternative to traditional numerical and semi‐analytical methods, aiding researchers and engineers working with SWCNTs and other nanomaterials.

Nonlocal buckling analysis of double-walled carbon nanotubes using initial value method and matricant approach

In this study, the buckling behavior of double-walled carbon nanotubes (DWCNTs) is investigated using the initial value method (IVM) combined with the approximate transfer matrix (ATM) approach within the framework of nonlocal elasticity theory. The aim is to provide a method that best estimates the critical buckling loads with improved accuracy and computational efficiency. To validate the method, the critical buckling loads obtained without considering nonlocal effects were compared with results from the literature, showing a high degree of agreement. Various boundary conditions, including simply supported (S-S), clamped-simply supported (C-S), clamped-free (C-F), and clamped-clamped (C-C), have been considered for investigation. The effects of nonlocal parameters, aspect ratios, and boundary conditions on the critical buckling loads are analyzed. The findings emphasize the importance of accounting for these parameters in mechanical analyses to ensure accurate predictions. The proposed framework provides an efficient alternative to existing numerical methods and offers valuable benchmarks for future research on the mechanical behavior of nanostructures.

The application of novel shear deformation theory and nonlocal elasticity theory to study the mechanical response of composite nanoplates

The use of composite structures, which have many layers of materials, has become more prevalent in the field of engineering. One of the advantages of this approach is its ability to use the inherent strengths of the constituent materials, resulting in a substantial increase in their load-bearing capability. Hence, this research represents the pioneering investigation into the static bending and free vibration characteristics of composite nanoplates including several layers of materials, whereby the material layers are interconnected via intricate profiles characterized by square wave and sine waveforms. The purpose of this endeavor is to fully capitalize on the benefits of attending courses in order to enhance practical working efficiency. This study also incorporates the use of two innovative third-order shear deformation theories. Simultaneously, considering the negligible size impact facilitated by the nonlocal theory, the mathematical formulations and equilibrium equations are derived using the Hamilton principle. The issue has been addressed using a four-node element with six degrees of freedom per node. One novel aspect of this study is its consideration of the impact of initial shape imperfections in various manifestations. Additionally, the elastic foundation incorporates characteristics that exhibit spatial variation. This statement provides a somewhat more accurate depiction of the behavior shown by actual structures. The numerical findings have been meticulously computed and thoroughly examined. Notably, it is possible to determine the optimal number of wavelengths in the profile to enhance the load-bearing capability of the structure. The findings derived from this study have significant value in informing the design of operational frameworks in practical settings.

Thermoelastic bending wave propagation of FG hybrid nanocomposite microbeam reinforced by GPLs and CNTs under fractional order nonlocal elasticity theory

With the size reduction of structures into micro/nanoscale, the size-dependent effect arises in mechanical properties. For micro/nanostructures working in transient thermal environment, the thermal relaxation needs to be considered in heat conduction. Functionally graded carbon nanotubes/graphene platelets reinforced composite (FG-CNTRC/GPLRC) materials as new generation of advanced materials have attracted much attention due to their excellent mechanical or thermoelastic performance. Though the mechanical or non-coupled thermoelastic behaviors of beam, plate and shell made of FG-CNTRC/GPLRC materials have been extensively studied, there lacks of investigations on the coupled thermoelastic behaviors of micro/nanostructures of FG-CNTRC/GPLRC in the context of generalized thermoelasticity where both the elastic nonlocality and thermal relaxation need to be considered. To fill this gap, in this work, the Lord–Shulman (L-S) type generalized thermoelasticity incorporating the fractional-order nonlocal theory is applied to investigating the thermoelastic bending wave propagation characteristics of a nanocomposite microbeam reinforced by GPLs and CNTs simultaneously for the first time. The Halpin–Tsai micromechanical model is used to assess the effective elastic modulus and the other properties are evaluated by the rule of mixture. Then, the governing equations are formulated based on the Euler–Bernoulli beam model. By assuming the wave-type solutions, the governing equations are solved and the dispersion relation between frequency and wave number and the relation between phase velocity and wave number are determined, respectively. In calculation, the effects of the thermal relaxation time, the elastic nonlocal parameter, the fractional order parameter and the mass fractions of GPLs and CNTs on the dimensionless frequency and phase velocity are analyzed and discussed in detail.

Thermal buckling of organic nanobeams resting on viscous elastic foundation

This study presents an analytical solution for organic solar nanobeams by including the new high-order shear deformation theory and considering the impact of size effects using non-local elasticity theory. The beam is supported by a viscoelastic base and experiences thermal loads that are distributed in accordance with uniform and nonlinear principles. The calculation formulae are derived from the consideration of significant deformations in the beam, resulting in increased complexity of the calculations. The equilibrium equation is derived from the fundamental principle of maximum work potential. The explicit equation for the critical thermal load is derived, providing a convenient means for the calculation and analysis of the impacts of relevant factors. The critical thermal buckling load encompasses both real and imaginary components, with the real component corresponding to the thermal load responsible for inducing beam instability, and the imaginary component associated with the dissipation of this load. This consideration is particularly pertinent when accounting for the foundation resistance. This finding enhances the level of interest in the research outcomes. This study also examined the impact of certain characteristic factors of the foundation and organic beams on the thermal buckling behavior of the beam, establishing a scientific basis for practical design applications.

Dynamic response of sandwich functionally graded nanoplate under thermal environments and elastic foundations using dynamic stiffness method

The present paper introduces the development of dynamic stiffness method for analyzing small-scale sandwich functionally graded nanoplates resting on elastic foundation in thermal environments. The mathematical formulation is based on classical plate theory in conjunction with nonlocal elasticity theory. The governing equation is derived using Hamilton’s principle. The dynamic stiffness matrix is obtained through the application of the Levy displacement approach and assembled to form the global stiffness matrix. The final matrix is solved for natural frequency of the plates using the Wittrick–Williams algorithm. The proposed methodology is validated against existing literature, demonstrating a strong agreement. Various parametric studies explore the effects of thermal environments, volume fraction index, sandwich configurations, elastic foundation characteristics, nonlocal parameter and boundary conditions. The results show the versatility of the proposed approach in addressing small scaled complex engineering structures. This research significantly contributes to the understanding and analysis of sandwich functionally graded nanoplates, providing valuable insights for applications in aerospace, structural systems, sensors, actuators, and energy harvesting devices.

Rayleigh waves in nonlocal orthotropic thermoelastic medium with diffusion

AbstractThis study examines Rayleigh wave propagation dynamics in a nonlocal orthotropic medium with thermoelastic diffusion, utilizing Eringen's nonlocal elasticity theory and the Lord–Shulman hyperbolic thermoelasticity model. Normal mode analysis is used to solve the problem, deriving the frequency equation for Rayleigh waves and analyzing specific cases. The elliptical path of surface particles and its eccentricity are calculated. Graphs illustrate the relationships of propagation speed, attenuation coefficient, penetration depth, and specific loss of Rayleigh waves concerning wave number for both thermally insulated and isothermal surfaces. The results reveal that the presence of nonlocal parameters and diffusion significantly increases the values of physical variables, especially with higher wave numbers, highlighting their considerable impact on the system's dynamics.

Thermomechanical interaction in a living tissue due to variable thermal loading with memory

AbstractIn order to address various clinical applications within living tissue, the aim of this work is to analytically study the thermomechanical interaction for a living tissue which is subjected to variable thermal loadings. Human tissues undergoing regional hyperthermia treatment for cancer therapy is based on graded changes of the cells, and as a consequences, the constitutive equations have been formulated using the nonlocal elasticity theory. The heat transport equation for the present problem is formulated in the context of Moore‐Gibson‐Thompson theory of generalized thermoelasticity assimilating the memory‐dependent derivative within a slipping interval. Both the boundaries of the tissue is maintaining the condition of zero traction. The lower boundary of the tissue is subjected to prescribed thermal loading while, the upper boundary is kept at zero temperature. Utilizing the Laplace transform mechanism, the governing equations have been solved and the general solutions have been obtained in the transformed domain. In order to arrive at the solutions in the real space‐time domain, suitable inversion of the Laplace transform has been carried out numerically using the method of Zakian. Numerical findings suggest that thermomechanical waves propagate through skin tissue over finite distances, which helps mitigate the unrealistic predictions made by the Pennes' model. Significant effect due to different effective parameter such as nonlocal parameter and the time‐delay parameter is reported. Also, how a nonlinear kernel function can be more effective in bio‐heat transfer, is outlined in the study also.

Eringen’s nonlocal elasticity theory for the analysis of two temperature generalized thermoelastic interactions in an anisotropic medium with memory

PurposeThe purpose of this paper is to investigate the thermoelastic interactions in a homogeneous, transversely isotropic infinite medium with a spherical cavity in the context of two temperature Lord-Shulman (2TLS) generalized theory of thermoelasticity considering Eringen’s nonlocal theory and memory dependent derivative (MDD). Memory-dependent derivative is found to be better than fractional calculus for reflecting the memory effect which leads us to the current investigation.Design/methodology/approachThe governing field equations of the problem are solved analytically using the eigenvalue approach in the transformed domain of Laplace when the cavity’s boundary is being loaded thermomechanically. Using MATLAB software the numerical solution in real space-time domain is obtained by Stehfest method.FindingsNumerical results for the different thermophysical quantities are presented in graphs and the effects of delay time parameter, non-local parameter and two temperature parameters are studied thereafter. The outcomes of this study convince that the displacement u, conductive temperature ϕ, thermodynamic temperature θ are concave upward whereas radial stress τrr is concave downward for every choice of delay time parameter ω, two temperature parameter η and non-local parameter “ζ”. As a specific instance of our findings, the conclusions of an equivalent problem involving integer order thermoelasticity theory can be obtained, and the corresponding results of this article can be readily inferred for isotropic materials.Originality/valueThe novelty of this research lies in the adoption of generalized thermoelastic theory with memory dependent derivative and Eringen’s nonlocality for analyzing the thermoelastic interactions in an infinite body with spherical cavity by employing eigenvalue approach. It has applications to many thermo-dynamical systems.

Thermo-elastic vibration analysis and optimization of a nanoresonator composed of a coupled nanotube system for acoustic liner application using the hybrid deep neural network–based white shark algorithm

This study explores the potential of nanoresonator systems composed of coupled nanotubes with graphene nanoparticles for enhancing the performance of acoustic liners in aircraft engines. The objective of this study is to develop an analytical model and predict its behavior under various conditions. The acoustic liners consist of perforated metal sheets and honeycomb cavities, which are essential for noise reduction in aircraft engines. The model is tested for variations in natural frequency, mode number (m), size effects (e0a), viscous constant (C), temperature (T), localness factor (L), and stiffness constant (K). A hybrid deep neural network–based white shark algorithm (DNN-WSA) is used to predict and optimize the performance of nanoresonator-coupled nanotube systems. Four theories were compared, such as wave propagation theory, nonlocal elasticity theory, polynomial eigenvalue approach, and governing equations with respect to natural frequencies in nanoresonator-coupled nanotube systems. The wave propagation theory yielded the lowest natural frequency, which was selected for detailed analysis. The optimized values of size effect of 2 nm, temperature of 5 K, and frequency of 1.971975 THz were obtained. When C = 0.3, K = 10, T = 300, and L = 10×e−9, the root mean square error (RMSE) value is 0.8421, which indicates improved predictive performance as it continues to decrease. The study’s findings showed that changes in viscous constants impact natural frequencies, while size effects have a minor influence. Temperature variations also affect natural frequencies, with higher temperatures leading to higher frequencies. The optimized model demonstrates enhanced predictive performance, which contributes to a better understanding of nanoresonator systems and their application in noise reduction for aircraft engines.

Size-dependent nonlinear free vibration of magneto-electro-elastic nanobeams by incorporating modified couple stress and nonlocal elasticity theory

Magneto-Electro-Elastic (MEE) Composites, as an innovative functional material blend, are composed of multiple materials, boasting exceptional strength, rigidity, and an extraordinary magneto-electric interaction effect. This paper establishes a nonlocal modified couple stress (NL-MCS) magneto-electro-elastic nanobeam dynamic model. To accurately capture the intricate influences of scale effects on nanostructures, This model meticulously examines scale effects from two distinct perspectives: leveraging nonlocal elasticity theory to elucidate the softening phenomena in nanostructures stemming from long-range particle interactions, and employing modified couple stress theory to reveal the hardening effects attributed to the rotational behavior of particles within the structure. By incorporating Von Karman geometric nonlinearity, Reddy’s third-order shear deformation theory and Maxwell’s equations, the governing equations for the nonlinear free vibration of MEE nanobeams are derived using Hamilton’s principle. Finally, a two-step perturbation method is employed to solve these equations. Two-step perturbation method disintegrates the solution process into two stages, iteratively approximating and refining the solution, thereby progressively unraveling the intricate details and enhancing the precision of the solution in a systematic manner. Finally, the nonlinear free vibration behavior of MEE nanobeams is explored under the coupled magnetic-electric-elastic fields, with a focus on the effects of various factors that including length scale parameters, nonlocal parameters, Winkler-Pasternak coefficients, span-to-thickness ratios, applied voltages and magnetic potentials.

Stability analysis of sandwich double nanobeam-system with varying cross-section interconnected by Kerr-type three-parameter elastic layer

In this study, the endurable buckling load of the elastically connected parallel sandwich nano-beams with varying cross-sections is assessed. In this regard, a layered beam system consisting of two parallel axially loaded tapered sandwich nanobeams that are interconnected via a Kerr-type three-parameter elastic foundation is considered. The geometric properties are assumed to be changed exponentially along the length of the beam element. The governing equilibrium equations of the system are described by a set of three coupled homogeneous differential equations, which originates in the context of Eringen's non-local elasticity theory and Euler beam model. Then, the numerical differential quadrature technique is used to estimate the endurable axial critical loads. Eventually, a comprehensive parameterization research is performed to investigate the sensitivity of linear buckling resistance to tapering ratio, nonlocal parameter, stiffness of elastic connections, volume fraction exponent, and thickness ratio. The research work of the present study is novel, and the attained numerical outcomes can be used as benchmarks for future researches in this field.

Investigation of vibration of nanorotating plates submerged in viscous-moving fluid medium

ABSTRACT In view of the great importance of dynamical behavior prediction of nanostructures in the fluid media and their various applications in biomedical engineering, aerospace, etc. in this research, the free vibration behavior of a nano-scale rotating plate coupled with an incompressible viscous fluid is studied. To this end, nonlocal elasticity theory is adopted to apply small-scale effects. Using the Navier–Stokes relation, the interaction forces between the fluid and nanorotating plate are obtained. In order to solve the governing differential equations, the Galerkin method is utilized, and the system’s vibration frequency response has been obtained for the clamped-free-free-free boundary condition.