• 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.

• Structural Engineering and Mechanics
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• v.69 no.2
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• pp.145-153
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• 2019

Modified couple stress formulation and first order shear deformation theory are used for magneto-electro-elastic bending analysis of three-layered curved size-dependent beam subjected to mechanical, magnetic and electrical loads. The governing equations are derived using a displacement field including radial and transverse displacements of middle surface and a rotation component. Size dependency is accounted based on modified couple stress theory by employing a small scale parameter. The numerical results are presented to study the influence of small scale parameter, initial electric and magnetic potentials and opening angle on the magneto-electro-elastic bending results of curved micro beam.

• 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.71 no.1
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• pp.37-43
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• 2019

This paper studies application of modified couple stress theory and first order shear deformation theory to magneto-electro-mechanical vibration analysis of three-layered size-dependent curved beam. The curved beam is resting on Pasternak's foundation and is subjected to mechanical, magnetic and electrical loads. Size dependency is accounted by employing a small scale parameter based on modified couple stress theory. The magneto-electro-mechanical preloads are accounted in governing equations to obtain natural frequencies in terms of initial magneto-electro-mechanical loads. The analytical approach is applied to investigate the effect of some important parameters such as opening angle, initial electric and magnetic potentials, small scale parameter, and some geometric dimensionless parameters and direct and shear parameters of elastic foundation on the magneto-electro-elastic vibration responses.

• Structural Engineering and Mechanics
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• v.65 no.1
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• pp.33-41
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• 2018

In this paper, using modified couple stress theory in place of classical continuum theory, and using shell model in place of beam model, vibrational behavior of nanotubes is investigated via the finite element method. Accordingly classical continuum theory is unable to correctly compute stiffness and account for size effects in micro/nanostructures, higher order continuum theories such as modified couple stress theory have taken on great appeal. In the present work the mass-stiffness matrix for cylindrical shell element is developed, and by means of size-dependent finite element formulation is extended to more precisely account for nanotube vibration. In addition to modified couple stress cylindrical shell element, the classical cylindrical shell element can also be defined by setting length scale parameter to zero in the equations. The boundary condition were assumed simply supported at both ends and it is shown that the natural frequency of nano-scale shell using the modified coupled stress theory is larger than that using the classical shell theory and the results of Ansys. The results have indicated using the modified couple stress cylindrical shell element, the rigidity of the nano-shell is greater than that in the classical continuum theory, which results in increase in natural frequencies. Besides, in addition to reducing the number of elements required, the use of this type of element also increases convergence speed and accuracy.

• Steel and Composite Structures
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• v.38 no.2
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• pp.213-221
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• 2021

The objective of this paper is to study the deformation in a homogeneous isotropic thermoelastic solid using modified couple stress theory subjected to ramp-type thermal source with two temperature. The advantage of this theory is the involvement of only one material length scale parameter which can determine the size effects. Laplace and Fourier transform technique is applied to obtain the solutions of the governing equations. The components of displacement, conductive temperature, stress components and couple stress are obtained in the transformed domain. A numerical inversion technique has been used to obtain the solutions in the physical domain. The effect of two temperature is depicted graphically on the resulted quantities. Numerical results show that the proposed model can capture the size effects of microstructures.

• Steel and Composite Structures
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• v.51 no.2
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• pp.103-116
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• 2024

Based on the consistent couple stress theory (CCST), we develop a unified formulation for analyzing the static bending and free vibration behaviors of functionally graded (FG) microscale beams (MBs). The strong forms of the CCST-based Euler-Bernoulli, Timoshenko, and Reddy beam theories, as well as the CCST-based sinusoidal, exponential, and hyperbolic shear deformation beam theories, can be obtained by assigning some specific shape functions of the shear deformations varying through the thickness direction of the FGMBs in the unified formulation. The above theories are thus included as special cases of the unified CCST. A comparative study between the results obtained using a variety of CCST-based beam theories and those obtained using their modified couple stress theory-based counterparts is carried out. The impacts of some essential factors on the deformation, stress, and natural frequency parameters of the FGMBs are examined, including the material length-scale parameter, the aspect ratio, and the material-property gradient index.

• Structural Engineering and Mechanics
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• v.46 no.3
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• pp.417-431
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• 2013

This paper presents responses of the free end of a cantilever micro beam under the effect of an impact force based on the modified couple stress theory. The beam is excited by a transverse triangular force impulse modulated by a harmonic motion. The Kelvin-Voigt model for the material of the beam is used. The considered problem is investigated within the Bernoulli-Euler beam theory by using energy based finite element method. The system of equations of motion is derived by using Lagrange's equations. The obtained system of linear differential equations is reduced to a linear algebraic equation system and solved in the time domain by using Newmark average acceleration method. In the study, the difference of the modified couple stress theory and the classical beam theory is investigated for the wave propagation. A few of the obtained results are compared with the previously published results. The influences of the material length scale parameter on the wave propagation are investigated in detail. It is clearly seen from the results that the classical beam theory based on the modified couple stress theory must be used instead of the classical theory for small values of beam height.

• Smart Structures and Systems
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• v.18 no.6
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• pp.1125-1143
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• 2016

Forced vibration analysis of a simple supported viscoelastic nanobeam is studied based on modified couple stress theory (MCST). The nanobeam is excited by a transverse triangular force impulse modulated by a harmonic motion. The elastic medium is considered as Winkler-Pasternak elastic foundation.The damping effect is considered by using the Kelvin-Voigt viscoelastic model. The inclusion of an additional material parameter enables the new beam model to capture the size effect. The new non-classical beam model reduces to the classical beam model when the length scale parameter is set to zero. The considered problem is investigated within the Timoshenko beam theory by using finite element method. The effects of the transverse shear deformation and rotary inertia are included according to the Timoshenko beam theory. The obtained system of differential equations is reduced to a linear algebraic equation system and solved in the time domain by using Newmark average acceleration method. Numerical results are presented to investigate the influences the material length scale parameter, the parameter of the elastic medium and aspect ratio on the dynamic response of the nanobeam. Also, the difference between the classical beam theory (CBT) and modified couple stress theory is investigated for forced vibration responses of nanobeams.

• Advances in nano research
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• v.6 no.3
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• pp.219-242
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• 2018

In this study, static bending of an edge cracked cantilever nanobeam composed of functionally graded material (FGM) subjected to transversal point load at the free end of the beam is investigated based on modified couple stress theory. Material properties of the beam change in the height direction according to exponential distributions. The cracked nanobeam is modelled using a proper modification of the classical cracked-beam theory consisting of two sub-nanobeams connected through a massless elastic rotational spring. The inclusion of an additional material parameter enables the new beam model to capture the size effect. The new non-classical beam model reduces to the classical beam model when the length scale parameter is set to zero. The considered problem is investigated within the Euler-Bernoulli beam theory by using finite element method. In order to establish the accuracy of the present formulation and results, the deflections are obtained, and compared with the published results available in the literature. Good agreement is observed. In the numerical study, the static deflections of the edge cracked FGM nanobeams are calculated and discussed for different crack positions, different lengths of the beam, different length scale parameter, different crack depths, and different material distributions. Also, the difference between the classical beam theory and modified couple stress theory is investigated for static bending of edge cracked FGM nanobeams. It is believed that the tabulated results will be a reference with which other researchers can compare their results.