Nonlocal vibration and buckling of two-dimensional layered quasicrystal nanoplates embedded in an elastic medium (original) (raw)
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Annals of Solid and Structural Mechanics, 2020
The focus of the present work is to present an analytical approach for buckling and free vibrations analysis of thick functionally graded nanoplates embedded in a Winkler-Pasternak medium. The equations of motion are derived according to both the third-order shear deformation theory, proposed by Reddy, and the nonlocal elasticity Eringen's model. For the first time, the equations are solved analytically for plates with two simply supported opposite edges, the solutions also turning helpful as shape functions in the analysis of structures with more complex geometries and boundary conditions. Sensitivity analyses are finally performed to highlight the role of nonlocal parameters, aspect and side-to-thickness ratios, boundary conditions, and functionally graded material properties in the overall response of plates and cylindrical shells. It is felt that the proposed strategy could be usefully adopted as benchmark solutions in numerical routines as well as for predicting some unexpected behaviors, for instance, in terms of buckling load, in thick nanoplates on elastic foundations.
Strojnícky časopis – Journal of MECHANICAL ENGINEERING, 2019
This paper shows an analysis of the free vibration of functionally graded simply supported nanoplate. The nonlocal four variables shear deformation plate theory is used to predict the free vibration frequencies of functionally graded nanoplate simply supported using non-local elasticity theory with the introduction of small-scale effects. The effect of the material properties, thickness-length ratio, aspect ratio, the exponent of the power law, the vibration mode is presented, the current solutions are compared to those obtained by other researchers. Equilibrium equations are obtained using the virtual displacements principle. P-FGM Power law is used to have a distribution of material properties that vary across the thickness. The results are in good agreement with those of the literature. 1 Introduction Functionally graded materials attract the attention of many researchers because of their powerful uses and their characteristics (mechanical, chemical, thermal, physical) such as high abrasion resistance (ceramic face), high impact resistance, reactor components, and insulating joints. Dimming improves the toughness of the ceramic face and prevents ceramic-metal detachment. Functionally graded materials (FGM) can be characterized by the gradual variation of material properties in the thickness. A new type of composite materials is developed recently (Abdelbaki et al [1]; Arnab Choudhury et al [2]; Abdelbaki et al [3]; Ebrahimi and Barati [4]; Ebrahimi and Heidari [5]; Elmerabet et al [6]; Elmossouess et al [7]; Houari et al [8]; Karami et al [9]; Mahjoobi and Bidgoli [10]; Mohamed et al [11]; Mokhtar et al [12]; Mokhtar et al [13]; Sadoun et al [14]; Salari et al [15]; Shafiei and Setoodeh [16]; Shokravi [17]; Tlidji et al [18]; Tounsi et al [19]; Tu et al [20]; Bocko, J et al [21]; Jozef, B et al [22]; Stephan, K et al [23]; Murín, J et al [24]; Sapountzakis, E et al [25]). Research work dealing with the behavior of nanoplates under different types of loading can be cited as Ansari and Norouzzadeh [26] studied the buckling responses of circular, elliptical and asymmetric nanometric plates in FGM. Banh-Thien et al [27] presented a new numerical approach for the buckling analysis of non-uniform thick nanoplates in an elastic medium using isogeometric analysis (IGA). Ghadiri et al [28] studied the vibrational frequency of orthotropic monolayer graphene sheets embedded in an elastic medium under the effect of the change of temperature, or the solution for the vibration of orthotropic rectangular nanoplates under thermal effect and the elastic medium is obtained with using GDQM. Liu et al [29] studied buckling and post-buckling behaviors of piezoelectric nanoplates subjected to combined thermo-electro-mechanical charges based on non-local theory, Mindlin's plate theory and von Karman's geometric nonlinearity. Arefi and Zenkour [30] presented the thermo-electro-magneto-elastic bending analysis of a three-layer sandwich nanoplate based on Unauthentifiziert | Heruntergeladen 11.12.19 17:52 UTC
2014
In this article, non-uniform biaxial buckling analysis of orthotropic single-layered graphene sheet embedded in a Pasternak elastic medium is investigated using the nonlocal Mindlin plate theory. All edges of the graphene sheet are subjected to linearly varying normal stresses. The nanoplate equilibrium equations are derived in terms of generalized displacements based on first-order shear deformation theory (FSDT) of orthotropic nanoplates using the nonlocal differential constitutive relations of Eringen. Differential quadrature method (DQM) has been used to solve the governing equations for various boundary conditions. The accuracy of the present results is validated by comparing the solutions with those reported by the available literatures. Finally, influences of small scale effect, aspect ratio, polymer matrix properties, type of planar loading, mode numbers and boundary conditions are discussed in details.
Journal of Intelligent Material Systems and Structures, 2017
This work is devoted to the free vibration nonlocal analysis of an elastic three-layered nanoplate with exponentially graded graphene sheet core and piezomagnetic face-sheets. The rectangular elastic three-layered nanoplate is resting on Pasternak’s foundation. Material properties of the core are supposed to vary along the thickness direction based on the exponential function. The governing equations of motion are derived from Hamilton’s principle based on first-order shear deformation theory. In addition, Eringen’s nonlocal piezo-magneto-elasticity theory is used to consider size effects. The analytical solution is presented to solve seven governing equations of motion using Navier’s solution. Eventually, the natural frequency is scrutinized for different side length ratio, nonlocal parameter, inhomogeneity parameter, and parameters of foundation numerically. The comparison with various references is performed for validation of our analytical results.
Vibration analysis of three-layered nanobeams based on nonlocal elasticity theory
Journal of Theoretical and Applied Mechanics, 2017
In this paper, the first investigation on free vibration analysis of three-layered nanobeams with the shear effect incorporated in the mid-layer based on the nonlocal theory and both Euler Bernoulli and Timoshenko beams theories is presented. Hamilton's formulation is applied to derive governing equations and edge conditions. In order to solve differential equations of motions and to determine natural frequencies of the proposed three-layered nanobeams with different boundary conditions, the generalized differential quadrature (GDQM) is used. The effect of the nanoscale parameter on the natural frequencies and deflection modes shapes of the three layered-nanobeams is discussed. It appears that the nonlocal effect is important for the natural frequencies of the nanobeams. The results can be pertinent to the design and application of MEMS and NEMS.
Journal of Nanomaterials, 2020
This article presents a finite element method (FEM) integrated with the nonlocal theory for analysis of the static bending and free vibration of the sandwich functionally graded (FG) nanoplates resting on the elastic foundation (EF). Material properties of nanoplates are assumed to vary through thickness following two types (Type A with homogeneous core and FG material for upper and lower layers and Type B with FG material core and homogeneous materials for upper and lower layers). In this study, the formulation of the four-node quadrilateral element based on the mixed interpolation of tensorial components (MITC4) is used to avoid “the shear-locking” problem. On the basis of Hamilton’s principle and the nonlocal theory, the governing equations for the sandwich FG nanoplates are derived. The results of the proposed model are compared with published works to verify the accuracy and reliability. Furthermore, the effects of geometric parameters and material properties on the static and ...
Composite Structures, 2019
This study is devoted to illustrate the mechanical buckling and free vibration analyses of double-porous functionally graded (FG) nanoplates embedded in an elastic foundation. A new quasi-3D refined plate theory is presented to model the displacement field. This theory contains only five unknown functions and considers the shear strain as well as thickness stretching. Based on the modified Mooney-type exponential relation, a new exponential law is presented to govern the materials variation and porosities distribution through the thickness of the nanoplates. The two porous nanoplates are bonded together by a set of parallel elastic springs and surrounded by Pasternak medium. The nonlocal strain gradient theory containing the nonlocal parameter and gradient coefficient is utilized to study the size-dependent effects. Based on Hamilton's principle, the equations of motion are drawn including the material parameters, elastic foundation reaction and biaxial compressive forces. An analytical approach for simply-supported and clamped bilayer porous FG nanoplates is implemented. The obtained results are compared with those available in the literature. Additional numerical calculations are introduced to show the influences of the material length scale parameters, inhomogeneity parameter, porosity factor and other parameters on the critical buckling and frequencies of the double-porous FG nanoplates.
Analysis and vibration of rectangular nanoplates - An overview
Egyptian Journal for Engineering Sciences and Technology, 2021
This work presents a review on solution methods and analysis of nanoplates structures with different boundary conditions and load cases, under some effects such as magnetic field and the effect of other parameters on the vibration and analysis. Moreover, it represents a review about the theories that are used to study these nanoplates structures such; the nonlocal elasticity theory by Eringen which is introduced to take into consideration the small scale effect of such nano-structure. The equation of motion of the nanoplate is derived then it used to study this type of nano-structures. Nanoplates are used in a lot of branches of life and in different applications because of the high and excellent mechanical, thermal and electrical properties of these nanoplates. Applications of nanoplate structure are also introduced to show the importance of studying and getting a solution of such nanoplates. Also, some parametric studies are discussed to show the effect of the studied parameters on the dynamic behaviour and analysis of these nanoplates.
Buckling and vibration analysis nanoplates with imperfections
Applied Mathematics and Computation, 2019
In the present paper a coupling finite strip-finite element procedure is developed to investigate the buckling and vibration behaviour of imperfect nanoplates via nonlocal Mindlin plate theory. The imperfection can be either a thickness variation or a lack of planarity and it can be either localized or distributed on an entire edge of the nanoplate. The resulting nonlinear equations are solved exactly by applying the Kantorovich method. A finite element approach is proposed for coupling the in-plane and the out-of-plane buckling equations to describe properly the imperfections. Some numerical examples are carried out in order to show the sensitivity of the results to the nonlocal parameter and to the imperfection.
Buckling analysis of variable thickness nanoplates using nonlocal continuum mechanics
Physica E: Low-dimensional Systems and Nanostructures, 2011
This paper presents an investigation on the buckling characteristics of nanoscale rectangular plates under bi-axial compression considering non-uniformity in the thickness. Based on the nonlocal continuum mechanics, governing differential equations are derived. Numerical solutions for the buckling loads are obtained using the Galerkin method. The present study shows that the buckling behaviors of single-layered graphene sheets (SLGSs) are strongly sensitive to the nonlocal and nonuniform parameters. The influence of percentage change of thickness on the stability of SLGSs is more significant in the strip-type nonoplates (nanoribbons) than in the square-type nanoplates.