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It has been revealed that application of the differential form of Eringen’s nonlocal elasticity theory to some cases (e.g. cantilevers) leads to paradoxical results, and recourse must be made to the integral version of Eringen’s nonlocal model. The purpose of this paper, within the framework of integral form of Eringen’s nonlocal theory, is to study the bending behavior of nanoscale plates with various boundary conditions using the isogeometric analysis (IGA).

The shear deformation effect is taken into account according to the Mindlin plate theory, and the minimum total potential energy principle is utilized in order to derive the governing equations. The relations are obtained in the matrix-vector form which can be easily employed in IGA or finite element analysis. For the comparison purpose, the governing equations are also derived based on the differential nonlocal model and are then solved via IGA. Comparisons are made between the predictions of integral nonlocal model, differential nonlocal model and local (classical) model.

The bending analysis of nanoplates under some kinds of edge supports indicates that using the differential model leads to paradoxical results (decreasing the maximum deflection with increasing the nonlocal parameter), whereas the results of integral model are consistent.

A new nonlocal formulation is developed for the IGA of Mindlin nanoplates. The nonlocal effects are captured based on the integral model of nonlocal elasticity. The formulation is developed in matrix-vector form which can be readily used in finite element method. Comparisons are made between the results of differential and integral models for the bending problem. The proposed integral model is capable of resolving the paradox appeared in the results of differential model.

This paper aims to present a new one-variable first-order shear deformation theory (OVFSDT) using nonlocal elasticity concepts for buckling of graphene sheets.

The FSDT had errors in its assumptions owing to the assumption of constant shear stress distribution along the thickness of the plate, even though by using the shear correction factor (SCF), it has been slightly corrected, the errors have been remained owing to the fact that the exact value of SCF has not already been accurately identified. By using two-variable first-order shear deformation theories, these errors decreased further by removing the SCF. To consider nanoscale effects on the plate, Eringen’s nonlocal elasticity theory was adopted. The critical buckling loads were computed by Navier’s approach. The obtained numerical results were then compared with previous studies’ results using molecular dynamics simulations and other plate theories for validation which also showed the accuracy and simplicity of the proposed theory.

In comparing the biaxial buckling results of the proposed theory with the two-variable shear deformation theories and exact results, it revealed that the two-variable plate theories were not appropriate for the investigation of asymmetrical analyses.

A formulation for FSDT was innovated by reconsidering its errors to improve the FSDT for investigation of mechanical behavior of nanoplates.

This paper aims to combine Eringen’s micromorphic and nonlocal theories and thus develop a comprehensive size-dependent beam model capable of capturing the effects of micro-rotational/stretch/shear degrees of freedom of material particles and nonlocality simultaneously.

To consider nonlocal influences, both integral (original) and differential versions of Eringen’s nonlocal theory are used. Accordingly, integral nonlocal-micromorphic and differential nonlocal-micromorphic beam models are formulated using matrix-vector relations, which are suitable for implementing in numerical approaches. A finite element (FE) formulation is also provided to solve the obtained equilibrium equations in the variational form. Timoshenko micro-/nano-beams with different boundary conditions are selected as the problem under study whose static bending is addressed.

It was shown that the paradox related to the clamped-free beam is resolved by the present integral nonlocal-micromorphic model. It was also indicated that the nonlocal effect captured by the integral model is more pronounced than that by its differential counterpart. Moreover, it was revealed that by the present approach, the softening and hardening effects, respectively, originated from the nonlocal and micromorphic theories can be considered simultaneously.

Developing a hybrid size-dependent Timoshenko beam model including micromorphic and nonlocal effects. Considering the nonlocal effect based on both Eringen’s integral and differential models proposing an FE approach to solve the bending problem, and resolving the paradox related to nanocantilever.

The purpose of this paper is concerned with the study of nonlinear ultrasonic waves in a magneto-flexo-thermo (MFT) elastic armchair single-walled carbon nanotube (ASWCNT) resting on polymer matrix.

A mathematical model is developed for the analytical study of nonlinear ultrasonic waves in a MFT elastic armchair single walled carbon nanotube rested on polymer matrix using Euler beam theory. The analytical formulation is developed based on Eringen’s nonlocal elasticity theory to account small scale effect. After developing the formal solution of the mathematical model consisting of partial differential equations, the frequency equations have been analysed numerically by using the nonlinear foundations supported by Winkler-Pasternak model. The solution is obtained by ultrasonic wave dispersion relations.

From the literature survey, it is evident that the analytical formulation of nonlinear ultrasonic waves in an MFT elastic ASWCNT embedded on polymer matrix is not discussed by any researchers. So, in this paper the analytical solutions of nonlinear ultrasonic waves in an MFT elastic ASWCNT embedded on polymer matrix are studied. Parametric studies is carried out to scrutinize the influence of the nonlocal scaling, magneto-electro-mechanical loadings, foundation parameters, various boundary condition and length on the dimensionless frequency of nanotube. It is noticed that the boundary conditions, nonlocal parameter and tube geometrical parameters have significant effects on dimensionless frequency of nanotubes.

This paper contributes the analytical model to find the solution of nonlinear ultrasonic waves in an MFT elastic ASWCNT embedded on polymer matrix. It is observed that the increase in the foundation constants raises the stiffness of the medium and the structure is able to attain higher frequency once the edge condition is C-C followed by S-S. Further, it is noticed that the natural frequency is arrived below 1% in both local and nonlocal boundary conditions in the presence of temperature coefficients. Also, it is found that the density and Poisson ratio variation affects the natural frequency with below 2%. The results presented in this study can provide mechanism for the study and design of the nano devices such as component of nano oscillators, micro wave absorbing, nano-electron technology and nano-electro--magneto-mechanical systems that make use of the wave propagation properties of ASWCNTs embedded on polymer matrix.

The purpose of this study is to analyze the thermo-diffusion process in a semi-infinite nonlocal fiber-reinforced double porous thermoelastic diffusive material with voids (FRDPTDMWV) in light of the fractional-order Lord–Shulman thermo-elasto-diffusion (LSTED) model. By virtue of Eringen’s nonlocal elasticity theory, the governing equations for the considered material are developed. The free surface of the substrate is governed by the inclined mechanical load and thermal and chemical shocks.

With the aid of the normal mode technique, the solutions of the nondimensional coupled governing equations have been obtained.

The expressions of field variables are obtained analytically. By using MATHEMATICA software, various graphical implementations are presented to describe the impacts of angle of inclination, fractional-order and nonlocality parameters. The present model is also validated on the basis of some comparative studies with some preestablished cases.

As observed from the literature survey, many different studies have been carried out by taking into account the deformation analysis in nonlocal double porous thermoelastic material structures and thermo-mechanical interaction in fiber-reinforced medium under fractional-order thermoelasticity theories. However, to the best of the authors’ knowledge, no research emphasizing the thermo-elasto-diffusive interactions in a nonlocal FRDPTDMWV has been carried out. Moreover, the effect of fractional-order LSTED theory on fiber-reinforced thermoelastic diffusive half-space with double porosity has not been illuminated till now, which significantly defines the novelty of the conducted research.

The aim of this paper is to study the dynamic vibrations of embedded double‐walled carbon nanotubes (DWCNTs) subjected to a moving harmonic load with simply supported boundary conditions.

The model of DWCNTs is considered as an Euler‐Bernoulli beam with waviness along the length, which is more accurate than the straight beam in previous works. Based on the nonlocal beam theory, the governing equations of motion are derived by using the Hamilton's principle, and then the separation of variables is carried out by the Galerkin approach, leading to two second‐order ordinary differential equations (ODEs).

The influences of the nonlocal parameter, the amplitude of the waviness, the surrounding elastic medium, the material length scale, load velocity and van der Waals force on the nonlinear vibration of DWCNTs are important.

The dynamic responses of DWCNTs are obtained by using the precise integrator method to ordinary differential equations.

The purpose of this article is to investigate the propagation characteristics (such as particle motion, attenuation and phase velocity) of a Rayleigh wave in a nonlocal generalized thermoelastic media.

The bulk waves are represented with Helmholtz potentials. The stress-free insulated and isothermal plane surfaces are taken into account. Rayleigh wave dispersion relation has been established and is found to be complex. Due to the presence of radicals, the dispersion equation is continuously computed as a complicated irrational expression. The dispersion equation is then converted into a polynomial equation that can be solved numerically for precise complex roots. The extra zeros in this polynomial equation are eliminated to yield the dispersion equation’s roots. These routes are then filtered for inhomogeneous wave propagation that decays with depth. To perform numerical computations, MATLAB software is used.

In this medium, only one mode of Rayleigh wave exists at both isothermal and insulated boundaries. The thermal factors of nonlocal generalized thermoelastic materials significantly influence the particle motion, attenuation and phase velocity of the Rayleigh wave.

Numerical examples are taken to examine how the thermal characteristics of materials affect the existing Rayleigh wave’s propagation characteristics. Graphical analysis is used to evaluate the behavior of particle motion (such as elliptical) both inside and at the isothermal (or insulated) flat surface of the medium under consideration.

In this paper, the important aspects of vibration analysis of carbon nanotubes (CNTs) as nano-resonators are studied. This study has covered the important nonlinear phenomena such as jump super-harmonic and chaotic behavior. CNT is modeled by using the modified nonlocal theory (MNT).

In previous research studies, the effects of CNT’s rotary inertia, stiffness and shear modulus of the medium were neglected. So by considering these terms in MNT, a comprehensive model of vibrational behavior of carbon nanotube as a nanosensor is presented. The nanotube is modeled as a nonlocal nonlinear beam. The first eigenmode of an undamped simply supported beam is used to extract the nonlinear equation of CNT. Harmonic balance method is used to solve the equation, while to study its super-harmonic behavior, higher-order harmonic terms were used.

In light of frequency response equation, jump phenomenon and chaotic behavior of the nanotube with respect to the amplitude of excitation are investigated. Also in each section of the study, the effects of elastic medium and nonlocal parameters on the vibration behavior of nanotube are investigated. Furthermore, parts of the results in linear and nonlinear cases were compared with results of other references.

The present modification of the nonlocal theory is so important and useful for accurate investigation of the vibrational behavior of nano structures such as a nano-resonator.

This paper aims to study photothermal excitation process in an initially stressed semi-infinite double porous thermoelastic semiconductor with voids subjected to Eringen’s nonlocal elasticity theory under the fractional order triple-phase-lag thermoelasticity theory. The considered substrate is governed by the mechanical and thermal loads at the free surface.

The normal mode technique is used to carry out the investigation of photothermal transportation. By virtue of the MATHEMATICA software, each distribution is exhibited graphically.

The expressions of the displacements, temperature, volume fractions of both kinds of voids, carrier density and stresses are determined analytically. With the help of the numerical data for silicon (Si) material, graphical implementations are presented on the basis of initial stress, fractional order, nonlocality and thermoelectric coupling parameters.

The present study fabricates the association of Eringen’s nonlocal theory and the stress analysis in a semiconducting double porous thermoelastic material with voids, which significantly implies the originality of the conducted work.

The purpose of the present paper is to investigate the thermo-mechanical interactions in an initially stressed nonlocal micropolar thermoelastic half-space having void pores under Lord–Shulman model. A moving thermal shock is applied to the formulation.

The normal mode technique is adopted to obtain the exact expressions of the physical quantities.

Numerical computations for stresses, displacement components, temperature field and change in the volume fraction field are performed for suitable material and are depicted graphically. Some comparisons have been shown in figures to estimate the effects of micropolarity, initial stress, voids, nonlocal parameter and time on the resulting quantities.

The exact expressions for the displacement components, stresses, temperature and change in the volume fraction field are obtained in the physical domain. Although numerous investigations do exist to observe the disturbances in a homogeneous, isotropic, initially stressed, micropolar thermoelastic half-space, the work in its current form has not been established by any scholar till now. The originality of the present work lies in the formulation of a fresh research problem to investigate the dependence of different physical fields on nonlocality parameters, micropolarity, initial stress, porosity and time due to the application of a moving thermal shock.