• Advances in materials Research
• /
• 제11권4호
• /
• pp.279-298
• /
• 2022

In this paper, the important novelty and the defining a physical phenomenon of the resent research is the development of nonlocal stress and strain parameters on the porous sandwich beam with functionally graded materials in the top and bottom face sheets.Also, various beam models including Euler-Bernoulli, Reddy and the generalized formulation of two-variable beam theories are obtained in this research. According to a nonlocal strain elasticity theory, the strain at a reference point in the body is dependent not only on the stress state at that point, but also on the stress state at all of the points throughout the body. Thus, the nonlocal stress-strain elasticity theory is defined that can be actual at micro/nano scales. It can be seen that the critical buckling load and transverse deflection of sandwich beam by considering both nonlocal stress-strain parameters is higher than the nonlocal stress parameter. On the other hands, it is noted that by considering the nonlocal stress-strain parameters simultaneously becomes the actual case.

• Structural Engineering and Mechanics
• /
• 제65권4호
• /
• pp.465-476
• /
• 2018

This article investigates buckling behavior of a multi-phase nanocrystalline nanobeam resting on Winkler-Pasternak foundation in the framework of nonlocal couple stress elasticity and a higher order refined beam model. In this model, the essential measures to describe the real material structure of nanocrystalline nanobeams and the size effects were incorporated. This non-classical nanobeam model contains couple stress effect to capture grains micro-rotations. Moreover, the nonlocal elasticity theory is employed to study the nonlocal and long-range interactions between the particles. The present model can degenerate into the classical model if the nonlocal parameter, and couple stress effects are omitted. Hamilton's principle is employed to derive the governing equations and the related boundary conditions which are solved applying an analytical approach. The buckling loads are compared with those of nonlocal couple stress-based beams. It is showed that buckling loads of a nanocrystalline nanobeam depend on the grain size, grain rotations, porosities, interface, elastic foundation, shear deformation, surface effect, nonlocality and boundary conditions.

• Advances in nano research
• /
• 제14권5호
• /
• pp.463-474
• /
• 2023

The purpose of this paper is to present the analysis of propagating and evanescent waves in functionally graded (FG) nanoplates with the consideration of nonlocal effect. The analytical integration nonlocal stress expansion Legendre polynomial method is proposed to obtain complete dispersion curves in the complex domain. Unlike the traditional Legendre polynomial method that expanded the displacement, the presented polynomial method avoids employing the relationship between local stress and nonlocal stress to construct boundary conditions. In addition, the analytical expressions of numerical integrations are presented to improve the computational efficiency. The nonlocal effect, inhomogeneity of medium and their interactions on wave propagation are studied. It is found that the nonlocal effect and inhomogeneity of medium reduce the frequency bandwidth of complex evanescent Lamb waves, and make complex evanescent Lamb waves have a higher phase velocity at low attenuation. The occurrence of intersections of propagating Lamb wave in the nonlocal homogeneous plate needs to satisfy a smaller Poisson's ratio condition than that in the classical elastic theory. In addition, the inhomogeneity of medium enhances the nonlocal effect. The conclusions obtained can be applied to the design and dynamic response evaluation of composite nanostructures.

• Advances in nano research
• /
• 제16권3호
• /
• pp.265-272
• /
• 2024

In this paper, the problem of static bending of axially functionally graded (AFG) nanobeam is formulated with the local stress(Lσ)- and strain-driven(εD) two-phase local/nonlocal integral models (TPNIMs). The novelty of the present study aims to compare the size-effects of nonlocal integral models on bending deflections of AFG Euler-Bernoulli nano-beams. The integral relation between strain and nonlocal stress components based on two types nonlocal integral models is transformed unitedly and equivalently into differential form with constitutive boundary conditions. Purely LσD- and εD-NIMs would lead to ill-posed mathematical formulation, and Purely εD- and LσD-nonlocal differential models (NDM) may result in inconsistent size-dependent bending responses. The general differential quadrature method is applied to obtain the numerical results for bending deflection and moment of AFG nanobeam subjected to different boundary and loading conditions. The influence of AFG index, nonlocal models, and nonlocal parameters on the bending deflections of AFG Euler-Bernoulli nanobeams is investigated numerically. A consistent softening effects can be obtained for both LσD- and εD-TPNIMs. The results from current work may provide useful guidelines for designing and optimizing AFG Euler-Bernoulli beam based nano instruments.

• Advances in nano research
• /
• 제8권3호
• /
• pp.191-202
• /
• 2020

The article brings the study of nonlocal, surface and the couple stress together to apparent the frequency retaliation of FG nanobeams (Functionally graded). For the examination of frequency retaliation, the article considers the accurate spot of neutral axis. This article aims to enhance the coherence of proposed model to accurately encapsulate the significant effects of the nonlocal stress field, size effects together with material length scale parameters. These considered parameters are assimilated through what are referred to as modified couple stress as well as nonlocal elasticity theories, which encompasses the stiffness-hardening and softening influence on the nanobeams frequency characteristics. Power-law distribution is followed by the functional gradation of the material across the beam width in the considered structure of the article. Following the well-known Hamilton's principle, fundamental basic equations alongside their correlated boundary conditions are solved analytically. Validation of the study is also done with published result. Distinct parameters (such as surface energy, slenderness ratio, as nonlocal material length scale and power-law exponent) influence is depicted graphically following the boundary conditions on non-dimensional FG nanobeams frequency.

• Interaction and multiscale mechanics
• /
• 제2권3호
• /
• pp.223-233
• /
• 2009

This paper presents a new nonlocal stress variational principle approach for the transverse free vibration of an Euler-Bernoulli cantilever nanobeam with an initial axial tension at its free end. The effects of a nanoscale at molecular level unavailable in classical mechanics are investigated and discussed. A sixth-order partial differential governing equation for transverse free vibration is derived via variational principle with nonlocal elastic stress field theory. Analytical solutions for natural frequencies and transverse vibration modes are determined by applying a numerical analysis. Examples conclude that nonlocal stress effect tends to significantly increase stiffness and natural frequencies of a nanobeam. The relationship between natural frequency and nanoscale is also presented and its significance on stiffness enhancement with respect to the classical elasticity theory is discussed in detail. The effect of an initial axial tension, which also tends to enhance the nanobeam stiffness, is also concluded. The model and approach show potential extension to studies in carbon nanotube and the new result is useful for future comparison.

• Smart Structures and Systems
• /
• 제21권3호
• /
• pp.279-286
• /
• 2018

The free longitudinal vibration of a circular truncated nanocone is investigated based on the nonlocal elasticity theory. Exact analytical formulations for tapered nanostructures are derived and the nonlinear differential governing equation of motion is developed. The nonlocal small scale effect unavailable in classical continuum theory is addressed to reveal the long-range interaction of atoms implicated in nonlocal constitutive relation. Unlike most previous studies applying the truncation method to the infinite higher-order differential equation, this paper aims to consider all higher-order terms to show the overall nonlocality. The explicit solution of nonlocal stress for longitudinal deformation is determined and it is an infinite series incorporating the classical stress derived in classical mechanics of materials and the infinite higher-order derivative of longitudinal displacement. Subsequently, the first three modes natural frequencies are calculated numerically and the significant effects of nonlocal small scale and vertex angle on natural frequencies are examined. The coupling phenomenon of natural frequency is observed and it is induced by the combined effects of nonlocal small scale and vertex angle. The critical value of nonlocal small scale is defined, and after that a new proposal for determining the range of nonlocal small scale is put forward since the principle of choosing the nonlocal small scale is still unclear at present. Additionally, two different types of nonlocal effects, namely the nonlocal stiffness weakening and strengthening, reversed phenomena existing in nanostructures are observed and verified. Hence the opposite nonlocal effects are resolved again clearly. The nano-engineers dealing with a circular truncated nanocone-based sensors and oscillators may benefit from the present work.

• Structural Engineering and Mechanics
• /
• 제9권2호
• /
• pp.209-214
• /
• 2000

The main purpose of this paper is to develop the results introduced in Artan (1996) and to find a general nonlocal linear elastic solution for Boussinesq problem. The general nonlocal solution given Artan (1996) is valid only when the distance to the boundary is greater than one atomic measure. The nonlocal stress field presented in this paper is valid for the whole half plane.

• Structural Engineering and Mechanics
• /
• 제66권2호
• /
• pp.237-248
• /
• 2018

This study presents the investigation of wave dispersion characteristics of a magneto-electro-elastic functionally graded (MEE-FG) nanosize beam utilizing nonlocal strain gradient theory (NSGT). In this theory, a material length scale parameter is propounded to show the influence of strain gradient stress field, and likewise, a nonlocal parameter is nominated to emphasize on the importance of elastic stress field effects. The material properties of heterogeneous nanobeam are supposed to vary smoothly through the thickness direction based on power-law form. Applying Hamilton's principle, the nonlocal governing equations of MEE-FG nanobeam are derived. Furthermore, to derive the wave frequency, phase velocity and escape frequency of MEE-FG nanobeam, an analytical solution is employed. The validation procedure is performed by comparing the results of present model with results exhibited by previous papers. Results are rendered in the framework of an exact parametric study by changing various parameters such as wave number, nonlocal parameter, length scale parameter, gradient index, magnetic potential and electric voltage to show their influence on the wave frequency, phase velocity and escape frequency of MEE-FG nanobeams.

• Structural Engineering and Mechanics
• /
• 제54권4호
• /
• pp.693-710
• /
• 2015

This paper presents a nonlocal shear deformation beam theory for bending, buckling, and vibration of functionally graded (FG) nanobeams using the nonlocal differential constitutive relations of Eringen. The developed theory account for higher-order variation of transverse shear strain through the depth of the nanobeam, and satisfy the stress-free boundary conditions on the top and bottom surfaces of the nanobeam. A shear correction factor, therefore, is not required. In addition, this nonlocal nanobeam model incorporates the length scale parameter which can capture the small scale effect and it has strong similarities with Euler-Bernoulli beam model in some aspects such as equations of motion, boundary conditions, and stress resultant expressions. The material properties of the FG nanobeam are assumed to vary in the thickness direction. The equations of motion are derived from Hamilton's principle. Analytical solutions are presented for a simply supported FG nanobeam, and the obtained results compare well with those predicted by the nonlocal Timoshenko beam theory.