• Steel and Composite Structures
• /
• v.35 no.1
• /
• pp.147-157
• /
• 2020

In the present research, thermo-elastic buckling of small scale functionally graded material (FGM) nano-size plates with clamped edge conditions rested on an elastic substrate exposed to uniformly, linearly and non-linearly temperature distributions has been investigated employing a secant function based refined theory. Material properties of the FGM nano-size plate have exponential gradation across the plate thickness. Using Hamilton's rule and non-local elasticity of Eringen, the non-local governing equations have been stablished in the context of refined four-unknown plate theory and then solved via an analytical method which captures clamped boundary conditions. Buckling results are provided to show the effects of different thermal loadings, non-locality, gradient index, shear deformation, aspect and length-to-thickness ratios on critical buckling temperature of clamped exponential graded nano-size plates.

• Advances in nano research
• /
• v.5 no.2
• /
• pp.113-126
• /
• 2017

This paper proposes a new nonlocal higher-order hyperbolic shear deformation beam theory (HSBT) for the static bending and vibration of nanoscale-beams. Eringen's nonlocal elasticity theory is incorporated, in order to capture small size effects. In the present model, the transverse shear stresses account for a hyperbolic distribution and satisfy the free-traction boundary conditions on the upper and bottom surfaces of the nanobeams without using shear correction factor. Employing Hamilton's principle, the nonlocal equations of motion are derived. The governing equations are solved analytically for the edges of the beam are simply supported, and the obtained results are compared, as possible, with the available solutions found in the literature. Furthermore, the influences of nonlocal coefficient, slenderness ratio on the static bending and dynamic responses of the nanobeam are examined.

• Advances in nano research
• /
• v.7 no.4
• /
• pp.265-275
• /
• 2019

In this work, the effect of size on the axial buckling behavior of single-layered graphene sheets embedded in elastic media is studied. We incorporate Eringen's nonlocal elasticity equations into three plate theories of first order shear deformation theory, higher order shear deformation theory, and classical plate theory. The surrounding elastic media are simulated using Pasternak and Winkler foundation models and their differences are evaluated. The results obtained from different nonlocal plate theories include the values of Winkler and Pasternak modulus parameters, mode numbers, nonlocal parameter, and side lengths of square SLGSs. We show here that axial buckling behavior strongly depends on modulus and nonlocal parameters, which have different values for different mode numbers and side lengths. In addition, we show that in different nonlocal plate theories, nonlocality is more influential in first order shear deformation theory, especially in certain range of nonlocal parameters.

• Structural Engineering and Mechanics
• /
• v.87 no.6
• /
• pp.529-540
• /
• 2023

A novel laminated-hybrid-composite-beam (LHCB) of glass-epoxy infused with flyash and graphene is constructed for this study. The conventional mixture-rule and constitutive-relationship are modified to incorporate filler and lamina orientation. Eringen's non-local-theory is used to include the filler effect. Hamilton's principle based on fifth-order-layer-wise-shear-deformation-theory is applied to formulate the equation of motion. The analogous shear-spring-models for LHCB with multiple-cracks are employed in finite-element-analysis (FEA). Modal-experimentations are conducted (B&K-analyser) and the findings are compared with theoretical and FEA results. In terms of dimensionless relative-natural-frequencies (RNF), the dynamic-response in cantilevered support is investigated for various relative-crack-severities (RCSs) and relative-crack-positions (RCPs). The increase of RCS increases local-flexibility in LHCB thus reductions in RNFs are observed. RCP is found to play an important role, cracks present near the end-support cause an abrupt drop in RNFs. Further, multiple cracks are observed to enhance the nonlinearity of LHCB strength. Introduction of the first to third crack in an intact LHCB results drop of RNFs by 8%, 10%, and 11.5% correspondingly. Also, it is demonstrated that the RNF varies because of the lamina-orientation, and filler addition. For 0° lamina-orientation the RNF is maximum. Similarly, it is studied that the addition of graphene reduces weight and increases the stiffness of LHCB in contrast to the addition of flyash. Additionally, the response of LHCB to moving mass is accessed by appropriately modifying the numerical programs, and it is noted that the successive introduction of the first to ninth crack results in an approximately 40% to 120% increase in the dynamic-amplitude-ratio.

• Advances in nano research
• /
• v.15 no.3
• /
• pp.215-224
• /
• 2023

Nonlinearity plays an important role in control systems and the application of design. For this reason, in addition to linear vibrations, nonlinear vibrations of the stepped nanobeam are also discussed in this manuscript. This study investigated the vibrations of stepped nanobeams according to Eringen's nonlocal elasticity theory. Eringen's nonlocal elasticity theory was used to capture the nanoscale effect. The nanoscale stepped Euler Bernoulli beam is considered. The equations of motion representing the motion of the beam are found by Hamilton's principle. The equations were subjected to nondimensionalization to make them independent of the dimensions and physical structure of the material. The equations of motion were found using the multi-time scale method, which is one of the approximate solution methods, perturbation methods. The first section of the series obtained from the perturbation solution represents a linear problem. The linear problem's natural frequencies are found for the simple-simple boundary condition. The second-order part of the perturbation solution is the nonlinear terms and is used as corrections to the linear problem. The system's amplitude and phase modulation equations are found in the results part of the problem. Nonlinear frequency-amplitude, and external frequency-amplitude relationships are discussed. The location of the step, the radius ratios of the steps, and the changes of the small-scale parameter of the theory were investigated and their effects on nonlinear vibrations under simple-simple boundary conditions were observed by making comparisons. The results are presented via tables and graphs. The current beam model can assist in designing and fabricating integrated such as nano-sensors and nano-actuators.

• Smart Structures and Systems
• /
• v.19 no.6
• /
• pp.601-614
• /
• 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
• /
• v.17 no.3
• /
• pp.275-284
• /
• 2024

In this article, a nonlocal stress-strain elasticity theory on the vibration analysis of Timoshenko sandwich beam theory with symmetric and asymmetric distributions of porous core and functionally graded material facesheets is introduced. According to nonlocal elasticity Eringen's theory (nonlocal stress elasticity theory), the stress at a reference point in the body is dependent not only on the strain state at that point, but also on the strain state at all of the points throughout the body; while, according to a new 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. Also, with combinations of two concepts, the nonlocal stress-strain elasticity theory is defined that can be actual at micro/nano scales. It is concluded that the natural frequency decreases with an increase in the nonlocal stress parameter; while, this effect is vice versa for nonlocal strain elasticity, because the stiffness of Timoshenko sandwich beam decreases with increasing of the nonlocal stress parameter; in which, the nonlocal strain parameter leads to increase the stiffness of structures at micro/nano scale. It is seen that the natural frequency by considering both nonlocal stress parameter and nonlocal strain parameter is higher than the nonlocal stress parameter only and lower for a nonlocal strain parameter only.

• Journal of the Korea Academia-Industrial cooperation Society
• /
• v.13 no.11
• /
• pp.5542-5550
• /
• 2012

Third-order shear deformation theory is reformulated using the nonlocal elasticity of Eringen. The equation of equilibrium of the nonlocal elasticity are derived. This theory has ability to capture the both small scale effects and quadratic variation of shear strain through the plate thickness. Navier's method has been used to solve the governing equations for all edges simply supported boundary conditions. Analytical solutions of buckling of nano-scale plates are presented using this theory to illustrate the effect of nonlocal theory on buckling load of the nano-scale plates. The relations between nonlocal third-order and local theories are discussed by numerical results. Further, effects of (i) length (ii) nonlocal parameter, (iii) aspect ratio and (iv) mode number on nondimensional buckling load are studied. In order to validate the present solutions, the reference solutions are used and discussed. The present results of nano-scale plates using the nonlocal theory can provide a useful benchmark to check the accuracy of related numerical solutions.

• Journal of the Korea Academia-Industrial cooperation Society
• /
• v.18 no.9
• /
• pp.52-60
• /
• 2017

This study examined the structural stability of nonlocal magneto-electro-elastic nano plates on elastic foundations using first-order shear deformation theory. Navier's method has been used to solve the buckling loads for all edges simply supported boundary conditions. On the other hand, biaxial buckling analysis of nano-plates has beenrarely studied. According to the Maxwell equation and the magneto-electro boundary condition, the change inthe magnetic and electric potential along the thickness direction of the magneto-electro-elastic nano plate wasdetermined. To reformulate the elasticity theory of the magneto- electro-elastic nano plate, the differential constitutive equation of Eringen was used and the governing equation of the nonlocal elasticity theory was studied using variational theory. The effects of the elastic foundation arebased on Pasternak's assumption. The relationship between nonlocal theory and local theory was analyzed through calculation results. In addition, structural stability problems were investigated according to the electric and magnetic potentials, nonlocal parameters, elastic foundation parameters, and side-to-thickness ratio. The results of the analysis revealedthe effects of the magnetic and electric potential. These calculations can be used to compare future research on new material structures made of magneto-electro-elastic materials.

• Structural Engineering and Mechanics
• /
• v.66 no.5
• /
• pp.621-629
• /
• 2018

By employing the nonlocal continuum field theory of Eringen and Von Karman nonlinear strains, this paper presents an analytical model for linear and nonlinear dynamics analysis of single-walled carbon nanotubes (SWNTs) conveying fluid with different boundary conditions. In the linear analysis the natural frequencies and critical flow velocities of SWNTs are computed. However, in the nonlinear analysis the effect of nonlocal parameter on nonlinear dynamics of cantilevered SWNTs conveying fluid is investigated by using bifurcation diagram, phase plane and Poincare map. Numerical results confirm existence of chaos as well as a period-doubling transition to chaos.