Size-Dependent Free Vibrations of FG Polymer Composite Curved Nanobeams Reinforced with Graphene Nanoplatelets Resting on Pasternak Foundations

Abstract

This paper presents a free vibration analysis of functionally graded (FG) polymer composite curved nanobeams reinforced with graphene nanoplatelets resting on a Pasternak foundation. The size-dependent governing equations of motion are derived by applying the Hamilton’s principle and the differential law consequent (but not equivalent) to Eringen’s strain-driven nonlocal integral elasticity model equipped with the special bi-exponential averaging kernel. The displacement field of the problem is here described in polar coordinates, according to the first order shear deformation theory. A large parametric investigation is performed, which includes different FG patterns, different boundary conditions, but also different geometrical parameters, number of layers, weight fractions, and Pasternak parameters.

Nanobeams with Internal Discontinuities: A Local/Nonlocal Approach

“…The behaviour of materials at the nanoscale level is significantly different from that exhibited by the same materials at the macroscale level [ 2 , 3 , 4 , 5 , 6 ]. In order to both analyse the so-called size effect and properly evaluate the size-dependent properties, two approaches may be followed: experimental characterisation [ 7 , 8 , 9 ] and theoretical modelling [ 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 ]. For instance, the tensile yield strength of a gradient nano-grained (GNG) surface layer in a bulk coarse-grained (CG) rod of a face-centred cubic Cu was investigated by Fang et al [ 7 ], who observed an increment of strength of about 100% with respect to that of a CG Cu.…”

“…However, despite their high level of reliability, performing experimental tests at the nanoscale may be quite expensive and time consuming, leading to often prefer theoretical models that are reliable and low-cost tool to estimate the behaviour of the nanomaterials. As a matter of fact, several theoretical models have been proposed aiming to capture the small-scale effect on the static [ 10 , 11 , 12 , 13 , 14 , 15 ] and dynamic responses [ 11 , 12 , 13 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 ], instability [ 12 , 24 , 25 , 26 , 27 , 28 ] and fracture behaviour [ 10 , 29 , 30 ] of nanomaterials at both the micro- or nano-scale level. In addition, the stress-driven nonlocal integral model (SDM) is available in the literature [ 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 ].…”

Nonlocal Elasticity Response of Doubly-Curved Nanoshells

“…A further application of the nonlocal elasticity theory can be found in Reference [23] for a parametric study of the axial post-buckling behavior of nanoshells with different nonlocal parameters. Various nonlocal theories have been applied within coupled problems, such as piezoelectric, flexoelectric, or thermo-electro-mechanical shells at different scales both for simple [24][25][26][27][28][29][30][31][32] or more complex [33][34][35][36][37][38][39][40][41][42][43][44][45][46] geometries.…”

A Modified Couple Stress Elasticity for Non-Uniform Composite Laminated Beams Based on the Ritz Formulation

“…Several experimental evidences in literature, have revealed that the behavior of micro-structures is size-dependent [2][3][4][5]. Thus, a large number of works has been recently published to conceive novel structural solutions, systems, and devices, while adopting different types of reinforcement phase, such as graphene nanoplatelets [6][7][8][9][10][11][12][13][14], or carbon nanotubes [15][16][17][18][19]. Among a large variety of numerical strategies, higher order theories represent the most useful tool for the investigation of the static and dynamic response of materials at different scales [20][21][22][23][24].…”