A novel one variable first-order shear deformation theory for biaxial buckling of a size-dependent plate based on the Eringen's nonlocal differential law.

Purpose – This paper aims to present a new one-variable first-order shear deformation theory (OVFSDT) using nonlocal elasticity concepts for buckling of graphene sheets. Design/methodology/approach – 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 resultswere 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. Findings – 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 a symmetrical analyses. Originality/value – A formulation for FSDT was innovated by reconsidering its errors to improve the FSDT for investigation of mechanical behavior of nanoplates.N/

On time-dependent nonlinear dynamic response of micro-elastic solids

A new approach to the mechanical response of micro-mechanic problems is presented using the modified couple stress theory. This model captured micro-turns due to micro-particles' rotations which could be essential for microstructural materials and/or at small scales. In a micro media based on the small rotations, sub-particles can also turn except the whole domain rotation. However, this framework is competent for a static medium. In terms of dynamic investigations of micro materials, it is required to involve micro-rotations' mass inertias. This fact persuades us to pay particular attention to the micro mechanics' samples and directed us to re-derive the modified couple stress model to propose and represent a new micro-mechanic approach which is well-deserved, especially for dynamic studies of microstructures. In carrying out this job, the classical beam has provided the basic form of formulation procedure. The continuum medium has been limited to a square flat non-porous beam deducing a homogeneous isotropic micromaterial. As long as the time-dependent results are concerned due to studying micro-mass inertia in time history, there would be two solution steps. The Galerkin decomposition technique is imposed in accord with an analytical postulate to issue the algebraic problem distributing time-dependent equations. The latter, the Homotopy perturbation method delivers time-dependent outcomes. The solution methods have been validated by building numerical models in Abaqus software. On the new achievements of this study, one can declare that both static and dynamic length scale parameters are very effective in order to study vibrations of microstructures. If the values of these characteristic lengths are considerable, the nonlinear frequency analysis will be essential. Furthermore, the stiffness of the structure will be higher if the values of both length scale parameters increase

On the Buckling Response of Axially Pressurized Nanotubes Based on a Novel Nonlocal Beam Theory

In the present study, the buckling analysis of single-walled carbon nanotubes (SWCNT) on the basis of a new refined beam theory is analyzed. The SWCNT is modeled as an elastic beam subjected to unidirectional compressive loads. To achieve this aim, the new proposed beam theory has only one unknown variable which leads to one equation similar to Euler beam theory and is also free from any shear correction factors. The equilibrium equation is formulated by the nonlocal elasticity theory in order to predict small-scale effects. The equation is solved by Navier’s approach by which critical buckling loads are obtained for simple boundary conditions. Finally, to approve the results of the new beam theory, some available well-known references are compared

Damped forced vibration analysis of single-walled carbon nanotubes resting on viscoelastic foundation in thermal environment using nonlocal strain gradient theory

In this paper, the damped forced vibration of single-walled carbon nanotubes (SWCNTs) is analyzed using a new shear deformation beam theory. The SWCNTs are modeled as a flexible beam on the viscoelastic foundation embedded in the thermal environment and subjected to a transverse dynamic load. The equilibrium equations are formulated by the new shear deformation beam theory which is accompanied with higher-order nonlocal strain gradient theory where the influences of both stress nonlocality and strain gradient size-dependent effects are taken into account. In this new shear deformation beam theory, there is no need to use any shear correction factor and also the number of unknown variables is the only one that is similar to the Euler-Bernoulli beam hypothesis. The governing equations are solved by utilizing an analytical approach by which the maximum dynamic deflection has been obtained with simple boundary conditions. To validate the results of the new proposed beam theory, the results in terms of natural frequencies are compared with the results from an available well-known reference. The effects of nonlocal parameter, half-wave length, damper, temperature and material variations on the dynamic vibration of the nanotubes, are discussed in detail. Keywords: Forced vibration, Single walled carbon nanotube, A new refined beam theory, Higher-order nonlocal strain gradient theory, Dynamic deflectio

Differential Quadrature Method for Dynamic Buckling of Graphene Sheet Coupled by a Viscoelastic Medium Using Neperian Frequency Based on Nonlocal Elasticity Theory

In the present study, the dynamic buckling of the graphene sheet coupled by a viscoelastic matrix was studied. In light of the simplicity of Eringen's non-local continuum theory to considering the nanoscale influences, this theory was employed. Equations of motion and boundary conditions were obtained using Mindlin plate theory by taking nonlinear strains of von Kármán and Hamilton's principle into account. On the other hand, a viscoelastic matrix was modeled as a three-parameter foundation. Furthermore, the differential quadrature method was applied by which the critical load was obtained. Finally, since there was no research available for the dynamic buckling of a nanoplate, the static buckling was taken into consideration to compare the results and explain some significant and novel findings. One of these results showed that for greater values of the nanoscale parameter, the small scale had further influences on the dynamic buckling

On a 3D material modelling of smart nanocomposite structures

Smart composites (SCs) are utilized in electro-mechanical systems such as actuators and energy harvesters. Typically, thin-walled components such as beams, plates, and shells are employed as structural elements to achieve the mechanical behavior desired in these composites. SCs exhibit various advanced properties, ranging from lower order phenomena like piezoelectricity and piezomagneticity, to higher order effects including flexoelectricity and flexomagneticity. The recently discovered flexomagneticity in smart composites has been investigated under limited conditions. A review of the existing literature indicates a lack of evaluation in three-dimensional (3D) elasticity analysis of SCs when the flexomagnetic effect (FM) exists. To address this issue, the governing equations will incorporate the term ∂/∂z, where z represents the thickness coordinate. The variational technique will guide us in further developing these governing equations. By using hypotheses and theories such as a 3D beam model, von Kármán's strain nonlinearity, Hamilton's principle, and well-established direct and converse FM models, we will derive the constitutive equations for a thick composite beam. Conducting a 3D analysis implies that the strain and strain gradient tensors must be expressed in 3D forms. The inclusion of the term ∂/∂z necessitates the construction of a different model. It should be noted that current commercial finite element codes are not equipped to accurately and adequately handle micro- and nano-sized solids, thus making it impractical to model a flexomagnetic composite structure using these programs. Therefore, we will transform the derived characteristic linear three-dimensional bending equations into a 3D semi-analytical Polynomial domain to obtain numerical results. This study demonstrates the importance of conducting 3D mechanical analyses to explore the coupling effects of multiple physical phenomena in smart structures

Electromagnetic forced vibrations of composite nanoplates using nonlocal strain gradient theory.

This article is intended to analyze forced vibrations of a piezoelectric-piezomagnetic ceramic nanoplate by a new refined shear deformation plate theory in conjunction with higher-order nonlocal strain gradient theory. As both stress nonlocality and strain gradient size-dependent effects are taken into account using the higher-order nonlocal strain gradient theory, the governing equations of the composite nanoplate are formulated. When the nanoplate is subjected to a transverse harmonic loading and all the edges are considered as simple boundaries, the governing equations can be solved with a closed-form solution, from which the maximum dynamic deflections are obtained. To validate the results of the new proposed plate theory, the comparisons between ours and the well-known papers in the literature are presented. The influences of different nonlocal parameters and material properties on the nanoplate's dynamic responses are also studied.N/

Buckling Analysis of a Micro Composite Plate with Nano Coating Based on the Modified Couple Stress Theory

The present study investigates the buckling of a thick sandwich plate under the biaxial non-uniform compression using the modified couple stress theory with various boundary conditions. For this purpose, the top and bottom faces are orthotropic graphene sheets and for the central core the isotropic soft materials are investigated. The simplified first order shear deformation theory (S-FSDT) is employed and the governing differential equations are obtained using the Hamilton’s principle by considering the Von-Karman’s nonlinear strains. An analytical approach is applied to obtain exact results with different boundary conditions. Due to the fact that there is no research on the stability of micro/nano sandwich plates based on S-FSDT including the couple stress effect, the obtained results are compared with the FSDT studies which use the Eringen nonlocal elasticity

Analytical predictions for the buckling of a nanoplate subjected to non-uniform compression based on the four-variable plate theory

In the present study, the buckling analysis of the rectangular nanoplate under biaxial non-uniform compression using the modified couple stress continuum theory with various boundary conditions has been considered. The simplified first order shear deformation theory (S-FSDT) has been employed and the governing differential equations have been obtained using the Hamilton’s principle. An analytical approach has been applied to obtain exact results from various boundary conditions. Due to the fact that there is not any research about the buckling of nanoplates based on the S-FSDT including the couple stress effect, the obtained results have been compared with the molecular dynamic simulation and FSDT papers which use the Eringen nonlocal elasticity theory. At the end, the results have been presented by making changes in some parameters such as the aspect ratio, the effect of various non-uniform loads and the length scale parameter

Analytical Buckling of FG Nanobeams on The Basis of A New One Variable First-Order Shear Deformation Beam Theory

In this work, buckling analysis of functionally graded (FG) nanobeams based on a new refined beam theory has been analyzed. The beam is modeled as an elastic beam subjected to unidirectional compressive loads. To achieve this aim, the new obtained beam theory has only one variable which lead to one equation similar to Euler beam theory and also is free of any shear correction factor. The equilibrium equation has been formulated by the nonlocal theory of Eringen to predict small-scale effects. The equation has been solved by Navier’s approach by which critical buckling loads have been obtained for simple boundaries. Finally, to approve the results of the new beam theory, various beam theories have been compared