3-D elasticity numerical solution for magneto-hygrothermal bending of FG graphene/metal circular and annular plates on an elastic medium
Introduction
Graphene exhibits exceptionally mechanical, physical and chemical properties (Potts et al., 2011), that consists a single layer of sp2 bonded carbon atoms organized in a 2-D hexagonal lattice. Graphene is the strongest material has been discovered so far (tensile strength equals about 130.5 GPa, Young's modulus is greater than 1 TPa and the mass of 1 m2 as 0.77 mg (Papageorgiou et al., 2017)). Moreover, the specific surface area of the graphene is 2630 m2/g (Papageorgiou et al., 2017), while it for carbon nanotubes is ranging from 100 to 1000 m2/g. A number of papers has been presented in the literature to elucidate the behavior of graphene and nanostructures (see e.g. (Sobhy, 2015; Sobhy and Zenkour, 2020; Sobhy and Zenkour, 2019a; Sobhy and Zenkour, 2018a; Radwan and Sobhy, 2020),). The composite reinforced materials have been widely used instead of the traditional materials in many significant industries such as aerospace, automotive, marine, etc. However, the properties of these reinforcement materials are somewhat limited comparing with the graphene platelets (GPLs). By comparing the carbon nanotube-reinforced composites ((Sobhy, 2019; Sobhy and Zenkour, 2018b; Sobhy and Radwan, 2020)) and the GPLs-reinforced composites, the one contains a high volume fraction of nanotube (about 60%) while the second contains a few weight fraction of graphene (0.5–20 % wt) (Bafana et al., 2017).
Various mechanical behaviors of GPLs-reinforced structures (beams, plates and shells) including analytical and numerical studies have been presented in the literature based on the different shear deformation theories (one- and two-dimensional theories) or the three-dimensional elasticity theory. Based on a one-dimensional higher-order shear deformation theory with thickness stretching effect, Polit et al. (2019) and (Anirudh et al., 2019) have investigated the bending, free vibration and buckling analyses of functionally graded (FG) GPLs curved beams with porosities employing the finite element method. The first-order shear deformation plate theory has been employed by Song et al. (2018) to investigate the bending and compressive buckling analyses of FG GPLs-reinforced plates using Navier's approach. Both first- and third-order shear deformation plate theories have been used by Li et al. (2018) to illustrate the bending, free vibration and buckling analyses of FG porous plates reinforced with GPLs. Sahmani et al. (2018) have employed the nonlocal strain gradient theory incorporating in the third-order shear deformable beam theory to investigate the nonlinear bending of porous micro/nano beams with GPLs reinforcements. Gholami and Ansari (2017) have investigated the nonlinear bending responses of the FG GPL-reinforced plates utilizing the sinusoidal shear deformation plate theory, the DQM, and the pseudo arc-length continuation method. Guo et al. (2018) have used the first-order shear deformation theory and the IMLS-Ritz approximation to illustrate the free vibration of GPLs-reinforced laminated quadrilateral plates. The uniaxial, biaxial and shear buckling and free vibration analyses of FG porous plates reinforced with GPLs have been presented by Yang et al. (2018) employing the first-order shear deformation theory and Chebyshev-Ritz method. The magneto-electro-thermal bending (Sobhy, 2018) and free vibration (Sobhy and Zenkour, 2019b) of FG GPLs-reinforced sandwich doubly-curved shallow shells have been studied by Sobhy (2018) and Sobhy and Zenkour (2019b), respectively, using the four-variable shear deformation shell theory. Effects of the magnetic field on the bending, buckling and free vibration of sandwich nanobeam with FG GPLs reinforced face layers have been investigated by Sobhy and his coworkers (Sobhy, 2020a, 2020b; Sobhy and Abazid, 2019; Zenkour and Sobhy, 2021) employing higher-order shear and normal deformation straight and curved beam theories. The buckling response of beams with GPLs reinforcements (Kitipornchai et al., 2017), buckling and postbuckling behaviors of FG GPLs-reinforced plates (Song et al., 2017) and wave dispersion in FG GPLs-reinforced nanoplates (Abazid et al., 1080) have been also investigated.
Annular and Circular structures are main contents utilized in several engineering applications such as aerospace, nuclear, mechanical, and civil industries. Safarpour et al. (2020) and Liu et al. (2019) have presented a semi-analytical solution for the bending and vibrational analysis of FG porous circular/annular plates (Safarpour et al., 2020) and FG annular plates (Liu et al., 2019) reinforced with GPLs under various boundary conditions using the state-space technique along the transverse direction and the DQM along the radial direction. Within the framework of 3-dimensional elasticity theory, thermoelastic bending analysis of FG GPLs-reinforced cylindrical shells with simply supported boundary conditions and exposed to uniform temperature rise has been presented by Alibeigloo (2020) using the state space approach. Yang et al. (2017a) have investigated the axisymmetric thermo-mechanical bending of circular and annular plates with GPLs reinforcements based on the 3D elasticity theory. While, in Yang et al. (2017b), thermoelastic bending response of FG GPLs-reinforced rectangular plates has been analyzed employing the 3D solution and the Mian and Spencer method.
As viewed in the above studies, the effects of the magnetic field, hygrothermal conditions and elastic foundations on the 3-D bending of circular/annular plates with FG GPLs reinforcements are not considered in the literature. Moreover, the numerical solution for the asymmetric and axisymmetric bending of circular/annular plates is presented for the first time by applying the DQM along both the radial and transverse directions. In addition, the 3D elasticity theory predicts more precise results than the 2D plate theories because it contains the lateral normal strain and the transverse normal stress.
The main object of this paper is to investigate the magneto-hygrothermal bending of FG GPLs-reinforced circular and annular plates embedded in an elastic medium exposed to in-plane magnetic field and elevated temperature in humid environment. The temperature and moisture are assumed to be varied sinusoidally along the radial and circumferential directions. While, they are obtained toward the thickness by solving the one-dimensional heat conduction and moisture diffusion equations. Young's modulus of GPLs is estimated according to the extended Halpin-Tsai model, while the other material properties are calculated based on a mixture law. All properties in terms of the GPLs volume fraction are graded based on a modified power rule. The three-dimensional governing equations are solved within the framework of the DQM that applied along the radial and transverse directions to obtain the stresses and displacements of the solid circular and annular plates. Appropriate convergence studies and comparison examples are introduced to verify the accuracy of the present results. Finally, some benchmark results are obtained.
Section snippets
Modeling the FG GPLs-reinforced circular/annular plates
Fig. 1 shows a circular/annular plate made of aluminum matrix reinforced with GPLs with inner radius , outer radius and thickness . The plate behavior is demonstrated based on the cylindrical coordinate system () that located at the center of the bottom surface. This model is subjected to transverse load applied to the top surface and rested on Pasternak elastic foundation at the bottom surface. The load-displacement relationship of the foundations is given as (Sobhy, 2019), (Sobhy
Constitutive equations
The constitutive equations for isotropic FG circular/annular plates considering the hygrothermal loads are expressed as:whereand the strain components () are related to the displacement field as:
Numerical solution
The governing equation (29) can be numerically solved by applying the DQM ((Al-Furjan et al., 2021a); (Al-Furjan et al., 2021b); (Al-Furjan et al., 2020a); (Al-Furjan et al., 2020b)) along both radial and thickness directions. On the basis of the DQM, the circular/annular plate can be discretized by mesh points along the radial direction (-axis) in the domain () and mesh points along the thickness direction (-axis) in the domain (). The and partial derivatives of a
Results and discussion
In the present analysis, the following dimensionless parameters are used:
The effects of various parameters on the asymmetric and axisymmetric bending of FG GPLs-reinforced circular/annular plates resting on elastic foundations and exposed to external mechanical load,
Conclusions
This article presents the 3-D asymmetric and axisymmetric bending response of FG GPLs/Al solid circular and annular plates under different boundary conditions. The volume fraction of graphene is varied through the thickness of the circular plate according to a modified power law. The circular and annular plates are rested on an elastic medium and exposed to transverse external load and hygrothermal loads as well as in-plane magnetic field. Based on the three-dimensional elasticity theory, the
Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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