• Multiscale and Multiphysics Mechanics
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• 제1권1호
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• pp.15-33
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• 2016

In this paper, an asymptotic method is employed to formulate nano- or micro-beams based on strain gradient elasticity. Although a basic theory for the strain gradient elasticity has been well established in literature, a systematic approach is relatively rare because of its complexity and ambiguity of higher-order elasticity coefficients. In order to systematically identify the strain gradient effect, an asymptotic approach is adopted by introducing the small parameter which represents the beam geometric slenderness and/or the internal atomistic characteristic. The approach allows us to systematically split the two-dimensional strain gradient elasticity into the microscopic one-dimensional through-the-thickness analysis and the macroscopic one-dimensional beam analysis. The first-order beam problem turns out to be different from the classical elasticity in terms of the bending stiffness, which comes from the through-the-thickness strain gradient effect. This subsequently affects the second-order transverse shear stress in which the surface shear stress exists. It is demonstrated that a careful derivation of a first strain gradient elasticity embraces "Gurtin-Murdoch traction" as the surface effect of a one-dimensional Euler-Bernoulli-like beam model.

• Structural Engineering and Mechanics
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• 제48권2호
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• pp.195-205
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• 2013

The buckling problem of linearly tapered micro-columns is investigated on the basis of modified strain gradient elasticity theory. Bernoulli-Euler beam theory is used to model the non-uniform micro column. Rayleigh-Ritz solution method is utilized to obtain the critical buckling loads of the tapered cantilever micro-columns for different taper ratios. Some comparative results for the cases of rectangular and circular cross-sections are presented in graphical and tabular form to show the differences between the results obtained by modified strain gradient elasticity theory and those achieved by modified couple stress and classical theories. From the results, it is observed that the differences between critical buckling loads achieved by classical and those predicted by non-classical theories are considerable for smaller values of the ratio of the micro-column thickness (or diameter) at its bottom end to the additional material length scale parameters and the differences also increase due to increasing of the taper ratio.

• Advances in nano research
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• 제16권2호
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• pp.175-186
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• 2024

Dynamical behaviors of one-dimensional (1D) nano-sized structures are of great importance in nanotechnology applications. Therefore, the torsional dynamic response of functionally graded nanorods which could be used to model the nano electromechanical systems or micro electromechanical systems with torsional motion about the center of twist is examined based on the theory of strain gradient nonlocal elasticity in this work. The mathematical background is constructed based on both strain gradient theory and Eringen's nonlocal elasticity theory. The equation of motions and boundary conditions of radially functionally graded nanorods are derived using Hamilton's principle and then transformed into the eigenvalue analysis by using Fourier sine series. A general coefficient matrix is obtained to assemble the Stokes' transformation. The case of a restrained functionally graded nanorod embedded in two elastic springs against torsional rotation is then deeply investigated. The effect of changing the functionally graded index, the stiffness of elastic boundary conditions, the length scale parameter and nonlocal parameter are investigated in detail.

• Advances in nano research
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• 제13권2호
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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.

• Advances in nano research
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• 제8권1호
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• pp.49-58
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• 2020

This paper investigates the vibration characteristics of flexoelectric nanobeams resting on viscoelastic foundation and subjected to magneto-electro-viscoelastic-hygro-thermal (MEVHT) loading. In this regard, the Nonlocal strain gradient elasticity theory (NSGET) is employed. The proposed formulation accommodates the nonlocal stress and strain gradient parameter along with the flexoelectric coefficient to accurately predict the frequencies. Further, with the aid of Hamilton's principle the governing differential equations are derived which are then solved through Galerkin-based approach. The variation of the natural frequency of MEVHT nanobeams under the influence of various parameters such as the nonlocal strain gradient parameter, different field loads, power-law exponent and slenderness ratio are also investigated.

• Advances in nano research
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• 제8권1호
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• pp.13-24
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• 2020

In this paper, for the first time based on the nonlocal strain gradient theory the effect of size dependency in torsional vibration of bi-direction functionally graded (FG) nonlinear nano-cone is study. The material properties were assumed to vary according to the arbitrary function in radial and axial directions. The Navier equation and boundary conditions of the size-dependent bidirectional FG nonlinear nano-cone were derived by Hamilton's principle. These equations were solved by employing the generalized differential quadrature method (GDQM). The presented model can turn into the classical model if the material length scale parameters are taken to be zero. The effects of some parameters, such as inhomogeneity constant, cross-sectional area parameter and small-scale parameters, were studied. As an essential result of this study can be stated that an FG nano-cone model based on the nonlocal elasticity theory behaves softer and based on the strain gradient theory behaves harder.

• Advances in nano research
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• 제14권5호
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• pp.475-493
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• 2023

In the context of nonclassical nonlocal strain gradient elasticity, this article studies the free and forced responses of functionally graded material (FGM) porous nanoplates exposed to thermal and magnetic fields under a moving load. The developed mathematical model includes shear deformation, size-scale, miscorstructure influences in the framework of higher order shear deformation theory (HSDT) and nonlocal strain gradient theory (NSGT), respectively. To explore the porosity effect, the study considers four different porosity models across the thickness: uniform, symmetrical, asymmetric bottom, and asymmetric top distributions. The system of quations of motion of the FGM porous nanoplate, including the effects of thermal load, Lorentz force, due to the magnetic field and moving load, are derived using the Hamilton's principle, and then solved analytically by employing the Navier method. For the free and forced responses of the nanoplate, the effects of nonlocal elasticity, strain gradient elasticity, temperature rise, magnetic field intensity, porosity volume fraction, and porosity distribution are analyzed. It is found that the forced vibrations of FGM porous nanoplates under thermal and live loads can be damped by applying a directed magnetic field.

• Steel and Composite Structures
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• 제28권1호
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• pp.13-24
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• 2018

A size-dependent novel hyperbolic shear deformation theory of simply supported functionally graded beams is presented in the frame work of the non-local strain gradient theory, in which the stress accounts for only the nonlocal strain gradients stress field. The thickness stretching effect (${\varepsilon}_z{\neq}0$) is also considered here. Elastic coefficients and length scale parameter are assumed to vary in the thickness direction of functionally graded beams according to power-law form. The governing equations are derived using the Hamilton principle. The closed-form solutions for exact critical buckling loads of nonlocal strain gradient functionally graded beams are obtained using Navier's method. The derived results are compared with those of strain gradient theory.

• Structural Engineering and Mechanics
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• 제48권3호
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• pp.415-434
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• 2013

This work is concerned with transverse vibrations of axially traveling nanobeams including strain gradient and thermal effects. The strain gradient elasticity theory and the temperature field are taken into consideration. A new higher-order differential equation of motion is derived from the variational principle and the corresponding higher-order non-classical boundary conditions including simple, clamped, cantilevered supports and their higher-order "offspring" are established. Effects of strain gradient nanoscale parameter, temperature change, shape parameter and axial traction on the natural frequencies are presented and discussed through some numerical examples. It is concluded that the factors mentioned above significantly influence the dynamic behaviors of an axially traveling nanobeam. In particular, the strain gradient effect tends to induce higher vibration frequencies as compared to an axially traveling macro beams based on the classical vibration theory without strain gradient effect.

• Steel and Composite Structures
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• 제27권2호
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• pp.201-216
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• 2018

In this paper, three-dimensional (3D) elasticity theory in conjunction with nonlocal strain gradient theory (NSGT) is developed for mechanical analysis of anisotropic nanoparticles. The present model incorporates two scale coefficients to examine the mechanical characteristics much accurately. All the elastic constants are considered and assumed to be the functions of (r, ${\theta}$, ${\varphi}$), so all kind of anisotropic structures can be modeled. Moreover, all types of functionally graded spherical structures can be investigated. To justify our model, our results for the radial vibration of spherical nanoparticles are compared with experimental results available in the literature and great agreement is achieved. Next, several examples of the radial vibration and wave propagation in spherical nanoparticles including nonlocal strain gradient parameters are presented for more than 10 different anisotropic nanoparticles. From the best knowledge of authors, it is the first time that 3D elasticity theory and NSGT are used together with no approximation to derive the governing equations in the spherical coordinate. Moreover, up to now, the NSGT has not been used for spherical anisotropic nanoparticles. It is also the first time that all the 36 elastic constants as functions of (r, ${\theta}$, ${\varphi}$) are considered for anisotropic and functionally graded nanostructures including size effects. According to the lack of any common approximations in the displacement field or in elastic constant, present theory can be assumed as a benchmark for future works.