• Advances in nano research
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• v.13 no.2
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• pp.147-164
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• 2022

This article investigates the longitudinal vibration of a nanorod embedded in viscoelastic medium according to the nonlocal strain gradient theory. Viscoelastic medium is considered based on Kelvin-Voigt model. Governing partial differential equation is derived based on longitudinal equilibrium and analytical solution is obtained by adopting harmonic motion solution for the nanorod. Modal frequencies and corresponding damping ratios are presented to demonstrate the influences of nonlocal parameter, material length scale, elastic and damping parameters of the viscoelastic medium. It is observed that material length scale parameter is very influential on modal frequencies especially at lower values of nonlocal parameter whereas increase in length scale parameter has less effect at higher values of nonlocal parameter when the medium is purely elastic. Elastic stiffness and damping coefficient of the medium have considerable impacts on modal frequencies and damping ratios, and the highest impact of these parameters on frequency and damping ratio is seen in the first mode. Results calculated based on strain gradient theory are quite different from those calculated based on classical elasticity theory. Hence, nonlocal strain gradient theory including length scale parameter can be used to get more accurate estimations of frequency response of nanorods embedded in viscoelastic medium.

• Steel and Composite Structures
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• v.22 no.6
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• pp.1239-1259
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• 2016

The modified couple stress-based third-order shear deformation theory is presented for sigmoid functionally graded materials (S-FGM) plates. The advantage of the modified couple stress theory is the involvement of only one material length scale parameter which causes to create symmetric couple stress tensor and to use it more easily. Analytical solution for dynamic instability analysis of S-FGM plates on elastic medium is investigated. The present models contain two-constituent material variation through the plate thickness. The equations of motion are derived from Hamilton's energy principle. The governing equations are then written in the form of Mathieu-Hill equations and then Bolotin's method is employed to determine the instability regions. The boundaries of the instability regions are represented in the dynamic load and excitation frequency plane. It is assumed that the elastic medium is modeled as Pasternak elastic medium. The effects of static and dynamic load, power law index, material length scale parameter, side-to-thickness ratio, and elastic medium parameter have been discussed. The width of the instability region for an S-FGM plate decreases with the decrease of material length scale parameter. The study is relevant to the dynamic simulation of micro structures embedded in elastic medium subjected to intense compression and tension.

• Steel and Composite Structures
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• v.26 no.1
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• pp.89-102
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• 2018

This work investigates a novel plate formulation and a modified couple stress theory that introduces a variable length scale parameter is presented to discuss the static and dynamic of functionally graded (FG) micro-plates. A new type of third-order shear deformation theory of Reddy that use only 4 unknowns by including undetermined integral variables is proposed in this study. The equations of motion are derived from Hamilton's principle. Analytical solutions are obtained for a simply supported micro-plate. Numerical examples are presented to examine the effect of the length scale parameter on the responses of micro-plates. The obtained results are compared with the previously published results to demonstrate the correctness of the present formulation.

• Structural Engineering and Mechanics
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• v.65 no.5
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• pp.573-585
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• 2018

This article presents strain gradient elasticity-based procedures for static bending, free vibration and buckling analyses of functionally graded rectangular micro-plates. The developed method allows consideration of smooth spatial variations of length scale parameters of strain gradient elasticity. Governing partial differential equations and boundary conditions are derived by following the variational approach and applying Hamilton's principle. Displacement field is expressed in a unified way to produce numerical results in accordance with Kirchhoff, Mindlin, and third order shear deformation theories. All material properties, including the length scale parameters, are assumed to be functions of the plate thickness coordinate in the derivations. Developed equations are solved numerically by means of differential quadrature method. Proposed procedures are verified through comparisons made to the results available in the literature for certain limiting cases. Further numerical results are provided to illustrate the effects of material and geometric parameters on bending, free vibrations, and buckling. The results generated by Kirchhoff and third order shear deformation theories are in very good agreement, whereas Mindlin plate theory slightly overestimates static deflection and underestimates natural frequency. A rise in the length scale parameter ratio, which identifies the degree of spatial variations, leads to a drop in dimensionless maximum deflection, and increases in dimensionless vibration frequency and buckling load. Size effect is shown to play a more significant role as the plate thickness becomes smaller compared to the length scale parameter. Numerical results indicate that consideration of length scale parameter variation is required for accurate modelling of graded rectangular micro-plates.

• Advances in nano research
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• v.6 no.2
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• pp.147-162
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• 2018

A new nonlocal higher order shear deformation theory (HSDT) is developed for buckling properties of single graphene sheet. The proposed nonlocal HSDT contains a new displacement field which incorporates undetermined integral terms and contains only two variables. The length scale parameter is considered in the present formulation by employing the nonlocal differential constitutive relations of Eringen. Closed-form solutions for critical buckling forces of the graphene sheets are obtained. Nonlocal elasticity theories are used to bring out the small scale influence on the critical buckling force of graphene sheets. Influences of length scale parameter, length, thickness of the graphene sheets and shear deformation on the critical buckling force have been examined.

• Structural Engineering and Mechanics
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• v.76 no.1
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• pp.27-56
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• 2020

In this paper, the deflection and buckling analyses of porous nano-composite piezoelectric plate reinforced by carbon nanotube (CNT) are studied. The equations of equilibrium using energy method are derived from principle of minimum total potential energy. In the research, the non-local strain gradient theory is employed to consider size dependent effect for porous nanocomposite piezoelectric plate. The effects of material length scale parameter, Eringen's nonlocal parameter, porosity coefficient and aspect ratio on the deflection and critical buckling load are investigated. The results indicate that the effect of porosity coefficient on the increase of the deflection and critical buckling load is greatly higher than the other parameters effect, and size effect including nonlocal parameter and the material length scale parameter have a lower effect on the deflection increase with respect to the porosity coefficient, respectively and vice versa for critical buckling load. Porous nanocomposites are used in various engineering fields such as aerospace, medical industries and water refinery.

• Steel and Composite Structures
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• v.39 no.5
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• pp.565-581
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• 2021

In this paper the buckling analysis of the nanoplate made of arbitrary bi-directional functionally graded (BDFG) materials with small scale effects are investigated. To study the small-scale effects on buckling load, the Eringen's nonlocal theory is applied. Employing the principle of minimum potential energy, the governing equations are obtained. Generalize differential quadrature method (GDQM) is used to solve the governing equations for various boundary conditions to obtain the buckling load of BDFG nanoplates. These models can degenerate into the classical models if the material length scale parameter is taken to be zero. Comparison between the results of GDQ method and other papers for buckling analysis of a simply supported rectangular nano FGM plate reveals the accuracy of GDQ method. At the end some numerical results are presented to study the effects of material length scale parameter, plate thickness, aspect ratio, Poisson's ratio boundary condition and side to thickness ratio on size dependent Frequency.

• Structural Engineering and Mechanics
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• v.68 no.1
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• pp.103-119
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• 2018

In the present research, an attempt is made to obtain a semi analytical solution for both nonlinear natural frequency and forced vibration of embedded functionally graded double layered nanoplates with all edges simply supported based on nonlocal strain gradient elasticity theory. The interaction of van der Waals forces between adjacent layers is included. For modeling surrounding elastic medium, the nonlinear Winkler-Pasternak foundation model is employed. The governing partial differential equations have been derived based on the Mindlin plate theory utilizing the von Karman strain-displacement relations. Subsequently, using the Galerkin method, the governing equations sets are reduced to nonlinear ordinary differential equations. The semi analytical solution of the nonlinear natural frequencies using the homotopy analysis method and the exact solution of the nonlinear forced vibration through the Harmonic Balance method are then established. The results show that the length scale parameters give nonlinearity of the hardening type in frequency response curve and the increase in material length scale parameter causes to increase in maximum response amplitude, whereas the increase in nonlocal parameter causes to decrease in maximum response amplitude. Increasing the material length scale parameter increases the width of unstable region in the frequency response curve.

• Structural Engineering and Mechanics
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• v.89 no.3
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• pp.225-238
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• 2024

In this paper, a quasi-3D hyperbolic shear deformation theory for the bending responses of a functionally graded (FG) porous micro-beam is based on a modified couple stress theory requiring only one material length scale parameter that can capture the size influence. The model proposed accounts for both shear and normal deformation effects through an illustrative variation of all displacements across the thickness and satisfies the zero traction boundary conditions on the top and bottom surfaces of the micro-beam. The effective material properties of the functionally graded micro-beam are assumed to vary in the thickness direction and are estimated using the homogenization method of power law distribution, which is modified to approximate the porous material properties with even and uneven distributions of porosity phases. The equilibrium equations are obtained using the virtual work principle and solved using Navier's technique. The validity of the derived formulation is established by comparing it with the ones available in the literature. Numerical examples are presented to investigate the influences of the power law index, material length scale parameter, beam thickness, and shear and normal deformation effects on the mechanical characteristics of the FG micro-beam. The results demonstrate that the inclusion of the size effects increases the microbeams stiffness, which consequently leads to a reduction in deflections. In contrast, the shear and normal deformation effects are just the opposite.

• Computers and Concrete
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• v.24 no.6
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• pp.499-511
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• 2019

In this paper, wave propagation of double-bonded Cooper-Naghdi micro sandwich cylindrical shells with porous core and carbon nanotube reinforced composite (CNTRC) face sheets are investigated subjected to multi-physical loadings with temperature dependent material properties. The governing equations of motion are derived by Hamilton's principle. Then, the influences of various parameters such as wave number, CNT volume fraction, temperature change, Skempton coefficient, material length scale parameter, porosity coefficient on the phase velocity of double-bonded micro sandwich shell are taken into account. It is seen that by increasing of Skempton coefficient, the phase velocity decreases for higher wave number and the results become approximately the constant. Also, by increasing of the material length scale parameter, the cut of frequency increases, because the stiffness of micro structure increases. The obtained results for this article can be used to detect, locate and quantify crack.