• Structural Engineering and Mechanics
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• v.54 no.4
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• pp.755-769
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• 2015

In this study, a 2-D finite element formulation in the frame of nonlocal integral elasticity is presented. Subsequently, the bending problem of a nanobeam under different types of loadings and boundary conditions is solved based on classical beam theory and also 3-D elasticity theory using nonlocal finite elements (NL-FEM). The obtained results are compared with the analytical and numerical results of nonlocal differential elasticity. It is concluded that the classical beam theory and the nonlocal differential elasticity can separately lead to significant errors for the problem under consideration as distinct from 3-D elasticity and nonlocal integral elasticity respectively.

• 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.

• Interaction and multiscale mechanics
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• v.3 no.3
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• pp.267-276
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• 2010

A nonlocal finite element model is developed for solving elasto-static frictional contact problems of nanostructures and nanoscale devices. A two dimensional Eringen-type nonlocal elasticity model is adopted. The material is characterized by a stress-strain constitutive relation of a convolution integral form whose kernel is capable to take into account both the diffusion process of nonlocal elasticity and the scale ratio effects. The incremental convex programming procedure is exploited as a solver. Two examples of different nature are presented, the first one presents the behavior of a nanoscale contacting system and the second example discusses the nano-indentation problem.

• Structural Engineering and Mechanics
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• v.72 no.2
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• pp.229-244
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• 2019

Weak form integral equations are developed to investigate the flapwise bending vibration of the rotating beams. Rayleigh and Eringen nonlocal elasticity theories are used to investigate the rotatory inertia and Size-dependency effects on the flapwise bending vibration of the rotating cantilever beams, respectively. Through repetitive integrations, the governing partial differential equations are converted into weak form integral equations. The novelty of the presented approach is the approximation of the mode shape function by a power series which converts the equations into solvable one. Substitution of the power series into weak form integral equations results in a system of linear algebraic equations. The natural frequencies are determined by calculation of the non-trivial solution for resulting system of equations. Accuracy of the proposed method is verified through several numerical examples, in which the influence of the geometry properties, rotatory inertia, rotational speed, taper ratio and size-dependency are investigated on the natural frequencies of the rotating beam. Application of the weak form integral equations has made the solution simpler and shorter in the mathematical process. Presented relations can be used to obtain a close-form solution for quick calculation of the first five natural frequencies of the beams with flapwise vibration and non-local effects. The analysis results are compared with those obtained from other available published references.

• Smart Structures and Systems
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• v.19 no.6
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• pp.601-614
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• 2017

In this work, free vibration analysis of size-dependent functionally graded (FG) nanoplates resting on two-parameter elastic foundation is investigated based on a novel nonlocal refined trigonometric shear deformation theory for the first time. This theory includes undetermined integral variables and contains only four unknowns, with is even less than the conventional first shear deformation theory (FSDT). Mori-Tanaka model is employed to describe gradually distribution of material properties along the plate thickness. Size-dependency of nanosize FG plate is captured via the nonlocal elasticity theory of Eringen. By implementing Hamilton's principle the equations of motion are obtained for a refined four-variable shear deformation plate theory and then solved analytically. To show the accuracy of the present theory, our research results in specific cases are compared with available results in the literature and a good agreement will be demonstrated. Finally, the influence of various parameters such as nonlocal parameter, power law indexes, elastic foundation parameters, aspect ratio, and the thickness ratio on the non-dimensional frequency of rectangular FG nanoscale plates are presented and discussed in detail.

• Advances in nano research
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• v.7 no.4
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• pp.277-292
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• 2019

In this present paper, a new two dimensional (2D) and quasi three dimensional (quasi-3D) nonlocal shear deformation theories are formulated for free vibration analysis of size-dependent functionally graded (FG) nanoplates. The developed theories is based on new description of displacement field which includes undetermined integral terms, the issues in using this new proposition are to reduce the number of unknowns and governing equations and exploring the effects of both thickness stretching and size-dependency on free vibration analysis of functionally graded (FG) nanoplates. The nonlocal elasticity theory of Eringen is adopted to study the size effects of FG nanoplates. Governing equations are derived from Hamilton's principle. By using Navier's method, analytical solutions for free vibration analysis are obtained through the results of eigenvalue problem. Several numerical examples are presented and compared with those predicted by other theories, to demonstrate the accuracy and efficiency of developed theories and to investigate the size effects on predicting fundamental frequencies of size-dependent functionally graded (FG) nanoplates.

• Steel and Composite Structures
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• v.34 no.5
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• pp.643-655
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• 2020

In the present work, the buckling behavior of a single-layered graphene sheet (SLGS) embedded in visco-Pasternak's medium is studied using nonlocal four-unknown integral model. This model has a displacement field with integral terms which includes the effect of transverse shear deformation without using shear correction factors. The visco-Pasternak's medium is introduced by considering the damping effect to the classical foundation model which modeled by the linear Winkler's coefficient and Pasternak's (shear) foundation coefficient. The SLGS under consideration is subjected to compressive in- plane edge loads per unit length. The influences of many parameters such as nonlocal parameter, geometric ratio, the visco-Pasternak's coefficients, damping parameter, and mode numbers on the buckling response of the SLGSs are studied and discussed.

• Structural Engineering and Mechanics
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• v.23 no.1
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• pp.63-74
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• 2006

In this paper, the scattering of harmonic elastic anti-plane shear waves by two collinear cracks in functionally graded materials is investigated by means of nonlocal theory. The traditional concepts of the non-local theory are extended to solve the fracture problem of functionally graded materials. To overcome the mathematical difficulties, a one-dimensional non-local kernel is used instead of a two-dimensional one for the anti-plane dynamic problem to obtain the stress field near the crack tips. To make the analysis tractable, it is assumed that the shear modulus and the material density vary exponentially with coordinate vertical to the crack. By use of the Fourier transform, the problem can be solved with the help of a pair of triple integral equations, in which the unknown variable is the displacement on the crack surfaces. To solve the triple integral equations, the displacement on the crack surfaces is expanded in a series of Jacobi polynomials. Unlike the classical elasticity solutions, it is found that no stress singularities are present at crack tips.

• Structural Engineering and Mechanics
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• v.64 no.4
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• pp.391-402
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• 2017

In this paper, a new nonlocal trigonometric shear deformation theory is proposed for thermal buckling response of nanosize functionally graded (FG) nano-plates resting on two-parameter elastic foundation under various types of thermal environments. This theory uses for the first time, undetermined integral variables and it contains only four unknowns, that is even less than the first shear deformation theory (FSDT). It is considered that the FG nano-plate is exposed to uniform, linear and sinusoidal temperature rises. Mori-Tanaka model is utilized to define the gradually variation of material properties along the plate thickness. Nonlocal elasticity theory of Eringen is employed to capture the size influences. Through the stationary potential energy the governing equations are derived for a refined nonlocal four-variable shear deformation plate theory and then solved analytically. A variety of examples is proposed to demonstrate the importance of elastic foundation parameters, various temperature fields, nonlocality, material composition, aspect and side-to-thickness ratios on critical stability temperatures of FG nano-plate.

• Smart Structures and Systems
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• v.21 no.1
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• pp.15-25
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• 2018

This work presents the buckling investigation of embedded orthotropic nanoplates such as graphene by employing a new refined plate theory and nonlocal small-scale effects. The elastic foundation is modeled as two-parameter Pasternak foundation. The proposed two-variable refined plate theory takes account of transverse shear influences and parabolic variation of the transverse shear strains within the thickness of the plate by introducing undetermined integral terms, hence it is unnecessary to use shear correction factors. Nonlocal governing equations for the single layered graphene sheet are obtained from the principle of virtual displacements. The proposed theory is compared with other plate theories. Analytical solutions for buckling loads are obtained for single-layered graphene sheets with isotropic and orthotropic properties. The results presented in this study may provide useful guidance for design of orthotropic graphene based nanodevices that make use of the buckling properties of orthotropic nanoplates.