Linear and nonlinear torsional free vibration of functionally graded micro/nano-tubes based on modified couple stress theory

Setoodeh, A.R.; Shahri, M. R. Zendehdel; Rezaei, Mohammad Parsa

doi:10.1007/s10483-016-2085-6

Cited by 21 publications

(9 citation statements)

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“…According to the modi ed couple stress theory, strain energy, U, for a deformable linear elastic material that occupies volume, V , can be written as follows [4]:…”

Section: Displacement Eld and Materials Propertiesmentioning

confidence: 99%

“…Thus, it is necessary to investigate the microstructure e ects on the mechanical behavior of the aforementioned structures. The local continuum theories could not capture the micro-sized e ects due to the lack of a material length-scale parameter; thus, non-classical theories, such as the couple stress, nonlocal elasticity, and strain gradient theories, have been proposed and widely implemented [2][3][4][5][6][7][8][9][10][11][12][13]. The strain gradient theory is a more general form of the higher-order non-classical theories taking into consideration both anti-symmetric and symmetric parts of the higher-order deformation gradients.…”

Section: Introductionmentioning

confidence: 99%

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An explicit solution for the size-dependent large amplitude transverse vibration of thin functionally graded micro-plates

Setoodeh

Rezaei

2017

Scientia Iranica

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Abstract. In this article, an analytical solution to the moderately large amplitude transverse vibration of thin Functionally Graded Micro-Plates (FGMPs) is presented based on a practical approach. The size-dependent nonlinear governing equation is obtained in conjunction with the Kirchho 's plate and modi ed couple stress theories. The material properties of Functionally Graded (FG) micro-plates vary according to the Reddy's model. The employed non-classical theory contains one material length-scale parameter to capture the size e ects. The highly nonlinear governing equation is solved by means of homotopy analysis method so as to obtain accurate analytic approximations. Both of simply supported and clamped micro-plates with immovable edges are considered. The comparison made between the present results and those of earlier studies con rms the reliability and e ectiveness of the present formulation for the design purpose. Furthermore, the e ects of di erent parameters, such as material gradient index, length-scale parameter, and aspect ratio, on the nonlinear frequency ratio are investigated.

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“…According to the modi ed couple stress theory, strain energy, U, for a deformable linear elastic material that occupies volume, V , can be written as follows [4]:…”

Section: Displacement Eld and Materials Propertiesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

An explicit solution for the size-dependent large amplitude transverse vibration of thin functionally graded micro-plates

Setoodeh

Rezaei

2017

Scientia Iranica

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“…(2020) presented effect of imperfect bonding on torsional vibrations of composite poroelastic cylinders; Yan et al. (2019) proposed a coupling atomistic-continuum model with Euler beam theory and capturing the size effect to investigate longitudinal and torsional free vibration characteristics of boron nitride nanotubes with both cantilevered and clamped constraints; Setoodeh et al. (2016) analytically studied linear and nonlinear torsional free vibration of functionally graded micro/nano-tubes based on modified couple stress theory; Yayli et al.…”

Section: Introductionmentioning

confidence: 99%

An indirect method to determine nonlinearly elastic shear stress-strain constitutive relationships for nonlinear torsional vibration of nonlinearly elastic shafts

Fan

Jian-she²,

Xue³

et al. 2022

MMMS

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PurposeIn this paper, the authors aim to propose an effective method to indirectly determine nonlinear elastic shear stress-strain constitutive relationships for nonlinear elasticity materials, and then study the nonlinear free torsional vibration of Al–1%Si shaft.Design/methodology/approachIn this study the authors use BoxLucas1 model to fit the determined-experimentally nonlinear elastic normal stress–strain constitutive relationship curve of Al–1%Si, a typical case of isotropic nonlinear elasticity materials, and then derive its nonlinear shear stress-strain constitutive relationships based on the fitting constitutive relationships and general equations of plane-stress and plane-strain transformation. Hamilton’s principle is utilized to gain nonlinear governing equation and boundary conditions for free torsional vibration of Al–1%Si shaft. Differential quadrature method and an iterative algorithm are employed to numerically solve the gained equations of motion.Findings The effect of four variables, namely dimensionless fundamental vibration amplitude ϑmax, radius α and length β, and nonlinear-elasticity intensity factor δ, on frequencies and mode shapes of the shafts is obtained. Numerical results are in good agreement with reference solutions, and show that compared with linearly elastic shear stress-strain constitutive relationships of the shafts made of the nonlinear elasticity materials, its actual nonlinearly elastic shear stress-strain constitutive relationships have smaller torsion frequencies. In addition, but β having opposite hardening effect, the rest of the four variables have softening effect on nonlinearly elastic torsion frequencies. Eventually, taking into account nonlinearly elastic shear stress-strain constitutive relationships, changes of the four factors, i.e. ϑmax, α, β and δ, cause inflation and deflation behaviors of mode shapes in nonlinear free torsional vibration.Originality/valueThe study could provide a reference for indirectly determining nonlinear elastic shear stress-strain constitutive relationships for nonlinear elasticity materials and for structure design of torsional shaft made of nonlinear elasticity materials.

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“…Setoodeh et al. 71 worked on linear and nonlinear FG micro/nanotubes based on the couple stress theory. The effects of the dimensions such as length scale, nonlocal parameter, volume fraction, responses amplitude on linear, and nonlinear fundamental frequencies were discussed, where there were two kinds of end supports.…”

Section: Introductionmentioning

confidence: 99%

Exponential and harmonic forced torsional vibration of single-walled carbon nanotube in an elastic medium

Khosravi

Hosseini

Norouzi

2020

Proceedings of the Institution of Mechanical Engineers, Part C:

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In this study, free torsional vibration and forced torsional vibration analysis under the time-dependent exponential and harmonic torsional loadings in single-walled carbon nanotube are investigated. The SWCNT is embedded in an elastic medium. Eringen's theory among the small-scale theories is selected. The nonlocal differential constitutive relation and corresponding boundary condition are derived via Hamilton's principle. Clamped–clamped boundary condition is utilized. The assumed modes method is employed for the dynamic torsional vibration in order to discretize the derived governing equations. The novelty of this work is devoted to the analysis of forced torsional vibration of a carbon nanotube embedded in an elastic medium under the various loadings. The angular displacement for the resonance frequency neglecting the elastic medium is illustrated. For the free analysis, the first three nondimensional natural frequencies with various small-scale parameters and stiffness of the elastic medium are calculated. The results are compared with another study for the first 10 mode numbers. The effects of the nonlocal parameter, length of carbon nanotube, stiffness of the elastic medium, thickness, time constant, and excitation frequency on the nondimensional and dimensional angular displacements are investigated, dynamically. For the greater values of the stiffness of the medium, the nonlocal parameter becomes negligible. When a time-dependent exponential torque is applied to the model, the angular displacement becomes greater and then lower by an increase in the value of the length, but the nondimensional angular displacement decreases continuously by increasing the value of the length under the time-dependent harmonic loading. Moreover, the angular displacement for a determined time becomes lower first and then becomes greater by increasing the time constant.

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Linear and nonlinear torsional free vibration of functionally graded micro/nano-tubes based on modified couple stress theory

Cited by 21 publications

References 1 publication

An explicit solution for the size-dependent large amplitude transverse vibration of thin functionally graded micro-plates

An explicit solution for the size-dependent large amplitude transverse vibration of thin functionally graded micro-plates

An indirect method to determine nonlinearly elastic shear stress-strain constitutive relationships for nonlinear torsional vibration of nonlinearly elastic shafts

Exponential and harmonic forced torsional vibration of single-walled carbon nanotube in an elastic medium

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