*Article* **Theoretical and Numerical Solution for the Bending and Frequency Response of Graphene Reinforced Nanocomposite Rectangular Plates**

**Mehran Safarpour 1, Ali Forooghi 1, Rossana Dimitri <sup>2</sup> and Francesco Tornabene 2,\***


**Abstract:** In this work, we study the vibration and bending response of functionally graded graphene platelets reinforced composite (FG-GPLRC) rectangular plates embedded on different substrates and thermal conditions. The governing equations of the problem along with boundary conditions are determined by employing the minimum total potential energy and Hamilton's principle, within a higher-order shear deformation theoretical setting. The problem is solved both theoretically and numerically by means of a Navier-type exact solution and a generalized differential quadrature (GDQ) method, respectively, whose results are successfully validated against the finite element predictions performed in the commercial COMSOL code, and similar outcomes available in the literature. A large parametric study is developed to check for the sensitivity of the response to different foundation properties, graphene platelets (GPL) distribution patterns, volume fractions of the reinforcing phase, as well as the surrounding environment and boundary conditions, with very interesting insights from a scientific and design standpoint.

**Keywords:** FG-GPL; GDQ; heat transfer equation; higher-order shear deformation theory

#### **1. Introduction**

Due to their outstanding thermal and mechanical properties, carbon-based nanofiller reinforced composites are widely applied in many engineering fields, such as civil, biomedical and automotive engineering [1–6]. In more detail, graphene platelets (GPLs) are increasingly introduced as carbon nano-fillers because of their relevant potentials in terms of high surface area, elasticity modulus, thermal conductivity, etc. GPLs, as one of novel nanosize reinforcements, have special properties, and their two-dimensional geometry enables them to be scattered in the matrix with less agglomeration, unlike the one-dimensional anisotropic ones. Due to their excellent mechanical, chemical, and physical properties, graphene-based composites demonstrate a wide range of applications in an engineering field, such as sensors, fuel cells, supercapacitors, and batteries. The addition of graphene as reinforcing agent in a polymer matrix, indeed, improves the overall performances and properties of composite materials, as largely demonstrated in the literature from researchers working in this area [7,8]. The primary interest of using graphene materials stems from their excellent mechanical, thermal, electrical and physicochemical properties with prosing results in all fields of technologies. For example, graphene represents one of the stiffest and most grounded materials, with an elastic modulus of ∼ 1 TPa and quality of ∼ 100 GPa [9–11]. By introducing 1 volume percent of graphene in a polymer matrix, the nanocomposite material reaches a conductivity of about 0.1 Sm−<sup>1</sup> with adequate consequences for electrical applications, along with significant changes in quality and strength [12]. In such a context, several theories and computational models have

**Citation:** Safarpour, M.; Forooghi, A.; Dimitri, R.; Tornabene, F. Theoretical and Numerical Solution for the Bending and Frequency Response of Graphene Reinforced Nanocomposite Rectangular Plates. *Appl. Sci.* **2021**, *11*, 6331. https://doi.org/10.3390/ app11146331

Academic Editor: Filippo Giannazzo

Received: 31 May 2021 Accepted: 6 July 2021 Published: 8 July 2021

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been developed in the last decades in the field of GPL-reinforced media. Anamagh and Bediz [13], for example, studied the buckling and vibration response of GPL-reinforced rectangular plates with different boundary conditions, based on a spectral-Tchebychev model. Reddy et al. [14] surveyed the vibrational frequencies of the composite plates reinforced by GPLs, and investigated the effect of various parameters, primarily, boundary conditions, distribution patterns, geometry and weight fractions of GPLs, on the natural frequencies of the system. In addition, Qaderi and Ebrahimi [15] focused on the frequency response of GPL reinforced rectangular plates embedded on viscoelastic substrates, and their sensitivity to different damping coefficients. In line with the previous works, Song et al. [16] studied the free and forced vibration behavior of FG-GPL-reinforced (FG-GPLR) plates by applying a first-order shear deformation theory (FSDT), with a clear enhancement of the vibration performances even with the addition of small quantities of GPLs. Based on the Chebyshev—Ritz procedure, Yang et al. [17] investigated the natural frequencies and critical buckling loads of FG-GPLR nanocomposite plates in presence of different porosities levels. Among the recent literature, different continuum-based nonlocal models have been considered as effective methods to treat plate-like nanostructures and to avoid possible difficulties encountered during experimental characterizations or time-consuming computational atomistic simulations of nanotubes. In this context, some theoretical studies of the free vibration response of graphene sheets can be found in the recent works [18–24], based on different nonlocal theoretical assumptions, accounting for different small-scale parameters, geometrical properties, boundary and environmental conditions. It is also well-known that different substrates can surround a structural member, thus affecting its mechanical behavior and stability. Numerous engineering problems (e.g., heavy machines, pavement of roads, etc.), indeed, are modeled as structural members resting on an elastic medium [25]. The elastic substrates are commonly modeled as Winkler or Pasternak foundations by means of one or two parameters [26,27]. The effect of visco-Pasternak substrate on the nonlinear dynamic response of the FG-GPLRC rectangular plates can be found in the seminal works by Fan et al. [28], and by Liu et al. [29] along with a sensitivity study of the mechanical behavior to different foundation parameters and porosity distributions. Among further works, Gao et al. [30] analyzed the nonlinear vibrational frequencies of FG-GPLR porous plates embedded on a two-parameter-type elastic medium, where an increased porosity coefficient was found to reduce the overall stiffness of structures. The vibrational properties of FG rectangular plates resting on a two-parameter elastic substrate were also surveyed by Thai and Choi [31]. They demonstrated that an increased quantity of metal components can significantly increase the deformability in a structural system. Similarly, Zhou et al. [32] studied the frequency response of thick plates on elastic media, while checking for the effect of different parameters, namely, the foundation coefficients, boundary conditions and aspect ratios, on the structural stiffness. A FSDT was also proposed in [33] to assess the nonlinear vibrational frequency and dynamic behavior of FG-GPLR plates resting on a viscoelastic-Pasternak foundation, with a clear reduction of the structural capacity for increased compressive loads.

Starting with the available literature on the topic, the present work aims at determining a general thermo-elasticity solution to treat both the static and frequency problems of GPLRC rectangular plates under different boundary conditions and embedding foundations, as typically applied in many lightweight mechanical and biomedical components, as well as in membranes and flexible wearable sensors and actuators. Despite the available literature on plate-like nanostructures, usually based on nonclassical approaches, the proposed work explores the capability of a higher-order shear deformation plate formulation combined with a modified Halpin and Tsai model to handle the problem, and checks for the potentials of the generalized differential quadrature (GDQ) approach as high-performance numerical tool to solve the equations even with a reduced computational effort, in lieu of the most common continuum finite element methods from the literature. The governing equations are here derived by means of the Hamilton's principle, accounting for a modified Halpin–Tsai model for the definition of the material properties and the effect

of the dispersion in nanocomposites. The GDQ-based solution is here compared to the analytical once based on a Navier-type expansion, and numerically. An extensive study is performed systematically to analyze the impact of different parameters such as the distribution patterns and weight fractions of the reinforcement phase, complex environments, Winkler–Pasternak foundation coefficients, and Kerr substrate constants on the overall response of FG-GPLRC rectangular plates. Results of the present study would be useful for the design of advanced lightweight composite members in civil and mechanical engineering, due to the importance of nanofillers dispersion and the application of foundation structures. The proposed GDQ method represents an innovative computational tool for design purposes, due to its great capability to solve challenging problems, with high simplicity and accuracy. A further extension of the formulation accounts for the thermal buckling of nanocomposite members within a unified setting, as useful for coupled problems for which theoretical predictions are usually cumbersome to obtain.

#### **2. Theoretical Formulation**

Here, we consider a FG-GPLRC rectangular plate resting on an elastic Winkler– Pasternak and Kerr medium, whose geometry and dimensions are depicted in Figure 1. The GPLs reinforcement is assumed to be distributed either uniformly (GPL-UD) or in a functionally graded way throughout the thickness, with two symmetric patterns, GPL-X, and GPL-O, respectively.

**Figure 1.** Rectangular plate embedded on and elastic foundation.

#### *2.1. Effective Material Properties*

The material properties are here defined according to a modified Halpin–Tsai model, such that the effective Young's modulus of the GPL/polymer composite *E* reads as follows [34]:

$$\overline{E} = \frac{3}{8} \frac{\left(1 + \xi\_L \eta\_L V\_{\rm GPL}\right)}{\left(1 - \eta\_L V\_{\rm GPL}\right)} E\_M + \frac{5}{8} \frac{\left(1 + \xi\_W \eta\_W V\_{\rm GPL}\right)}{\left(1 - \eta\_W V\_{\rm GPL}\right)} E\_M \tag{1}$$

where

$$\mathfrak{z}\_{L}^{\mathfrak{x}} = 2 \frac{L\_{\mathrm{GPL}}}{t\_{\mathrm{GPL}}}, \mathfrak{z}\_{W}^{\mathfrak{x}} = 2 \frac{W\_{\mathrm{GPL}}}{t\_{\mathrm{GPL}}}, \eta\_{W} = -\frac{1 - \left(\frac{E\_{\mathrm{GPL}}}{E\_{M}}\right)}{\mathfrak{z}\_{W} + \left(\frac{E\_{\mathrm{GPL}}}{E\_{M}}\right)}, \eta\_{L} = \frac{\left(\frac{E\_{\mathrm{GPL}}}{E\_{M}}\right) - 1}{\mathfrak{z}\_{L} + \left(\frac{E\_{\mathrm{GPL}}}{E\_{M}}\right)}\tag{2}$$

with *EM* and *EGPL* are the Young's moduli of the polymer matrix and GPLs, respectively; *VGPL* is the GPL volume fraction, *ξ<sup>L</sup>* and *ξ<sup>W</sup>* are the parameters characterizing both the geometry and size of GPL nanofillers; *LGPL*, *WGPL* and *tGPL* are the average length, width, and thickness of GPLs, respectively.

In line with findings by Rafiee et al. [35], the effective Young's modulus of GPL/polymer nanocomposites is well-approximated by the modified Halpin–Tsai model. The result determined by Equation (1), indeed, is only 2.7% higher than the experimental predictions. Based on the same rule of mixtures, the effective Poisson's ratio and mass density read as follows:

$$
\overline{\rho} = \rho\_{\rm GPL} V\_{\rm GPL} + \rho\_M (1 - V\_{\rm GPL}), \quad \overline{\nu} = \nu\_{\rm GPL} V\_{\rm GPL} + \nu\_M (1 - V\_{\rm GPL}) \tag{3}
$$

while, the effective shear modulus is defined as:

$$\overline{G} = \frac{\overline{E}}{2(1+\mathbb{F})} \tag{4}$$

As also depicted in Figure 2, we select three different distribution patterns of GPLs along the thickness direction of the structure, whose analytical expressions take the following form [36]:

**Figure 2.** Distribution patterns of GPLs: (**a**) GPL-UD distribution, (**b**) GPL-X distribution, (**c**) GPL-O distribution.
