Bending analysis of functionally graded graphene oxide powder-reinforced composite beams using a meshfree method

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Abstract

In this paper, the linear bending response of straight, thin to moderately thick functionally graded (FG) composite beams reinforced with graphene oxide powder (GOP) and subjected to two types of static mechanical loads, with various boundary conditions is investigated using a meshfree collocation method based on multiquadric radial basis functions. The weight fractions of the GOP are considered to vary continuously through the thickness direction of the composite beams. To determine the effective Young's modulus of the FG-GOP reinforced composite beams, a modified Halpin–Tsai model is used, whereas the rule of mixture is adopted to estimate the equivalent Poisson's ratio. The first-order shear deformation (FSDT) beam theory is adopted to model the functionally graded graphene oxide powder (FG-GOP) reinforced composite beams. The static equilibrium equations and associated boundary conditions of the FG-GOP reinforced composite beams are obtained by using the principle of minimum potential energy. The governing equations are rewritten in a matrix-vector strong form and discretized using the multiquadric radial basis functions. Various comparisons studies are performed to demonstrate the robustness and accuracy of the proposed numerical model. Parametric studies are conducted to examine the effects of length-to-thickness ratio, GOP distribution patterns, weight fraction, and size of GOP, types of loads, and boundary conditions on the bending behavior of the FG-GOP reinforced composite beams.

Introduction

In recent years, the use of composite materials has become important in many fields of engineering [1] owing to their marvelous characteristics [2] compared to conventional materials. A composite material which may be either natural or synthetic also termed for briefness in materials science, a composite, is an advanced material that is fabricated to fulfill a specific structural function. Composite materials are the results of combining on a macroscopic scale of at least two different materials that have dissimilar physical and chemical properties. The remarkable advantage of composite materials is that they possess properties that are not available and cannot be provided by individual components alone. In general, composite materials are composed of three different phases, the first one is a discontinuous phase termed the reinforcement or fiber, the second one is a continuous phase called the matrix which may be a polymer, metal, or ceramic [3], and the third one is the interfacial region between the reinforcing and matrix phases. Reinforcements can be of different nature, shape, and size. It is known that the reinforcement component, plays an interesting role in the improvement of composite's properties through providing weight saving, strength, and stiffness. Therefore, it is of the utmost importance to choose carefully the reinforcement for an optimal fabrication of composite materials.

One of the most frequently used reinforcing materials are carbon nanotubes (CNTs) which are promising nanomaterials that have gained the attention of academic and industrial communities due to their exceptional and useful properties [4]. Several studies have been made to analyze the behavior of composite structures reinforced with CNT. Bourada et al. [5] have used an integral first-order shear deformation beam theory for the purpose of investigating the dynamic and stability responses of a simply supported single walled carbon nanotubes reinforced concrete beam on elastic foundation. Bousahla et al. [6] presented a novel integral first order shear deformation theory for the buckling and dynamic behaviors of simply supported CNT-RC composite beams. The characteristics of waves propagated in a nanocomposite sandwich panel are studied [7]. The nonlinear frequency and chaotic behaviors of the multi-scale hybrid nano-composite reinforced disk in the thermal environment were investigated using the generalized differential quadrature method [8]. In [9], the finite element method was adopted for nonlinear analysis of viscoelastic micro-composite beam with consideration of geometrical imperfection. Bending response of FG-CNT reinforced sandwich plates was investigated [10], [11]. The frequency response of imperfect honeycomb core sandwich disk with multiscale hybrid nanocomposite (MHC) face sheets rested on an elastic foundation was analyzed [12]. The free vibration behavior of a sandwich disk with a lactic core (honeycomb), and two middle layers containing SMA fiber was examined [13]. The nonlinear thermal instability response of functionally graded graphene nanoplatelet-reinforced composite (FG-GPLRC) disk was investigated using the generalized differential quadrature method [14]. In addition to CNTs, it should be noted that there are some important nanofillers that are consisted of other carbon-based materials and can be utilized as alternative reinforcement components in the design of composite structures. Common examples of these substances are graphene platelets (GPLs) and graphene oxide powders (GOPs) [15]. The GOP is considered a novel nanofiller that possess interesting and attractive mechanical, thermal, and optical properties [16]. Recently, the GOP has been found to be in remarkable compatibility with polymers.

The functionally graded materials (FGMs) [17] are a modern class of composites in which both composition and material properties varying continuously as a function of position over a certain direction(s) of the structure, and, hence, permit to avoid some common problems observed in conventional composites [18]. The bending response of functionally graded porous plates is studied using a cubic shear deformation theory [19]. The buckling and bending analyses of porous functionally graded (FG) plate under mechanical load are presented using refined trigonometric shear deformation theory [20]. Cuong et al. [21] have used the Isogeometric method to investigate vibration and buckling responses of annular plates, conical, and cylindrical shells made of functionally graded porous materials. A new higher-order shear deformation theory was proposed by Rabhi et al. [22] to investigate the buckling and free vibration behaviors of graded sandwich plates resting on elastic foundations. Menasria et al. [23] have proposed a new four-unknown refined plate theory to study the dynamic response of FG-sandwich plates. The geometrically nonlinear static behavior of porous functionally graded (PFG) micro-plate was studied [24]. Using the Rayleigh-Ritz technique, the rotating vibration behavior of functionally graded shell with ring supports is examined [25]. A new refined trigonometric shear deformation theory in conjunction with Galerkin's was used in the analysis of the buckling response of FG sandwich plates [26]. The linear stability behavior of FG sandwich plates resting on Winkler-Pasternak elastic foundation and subjected to hygro-thermal-mechanical loads was analyzed using a novel shear deformation theory [27]. The functionally graded GOP reinforced composite material in which the GOP reinforcement is supposed dispersed uniformly or nonuniformly within the matrix phase is a new advanced composite material with promising characteristics since it combines the advantages of both FGMs and GOP. In these recent years, there is an important effort to analyze and design structural components fabricated from composites. Functionally graded graphene oxide powder FG-GOP reinforced composite beams are a type of these ideal and new structural composite elements. For an efficient maintenance and high quality manufacturing of FG-GOP reinforced composite beams, it is of paramount importance to explore their mechanical behaviors under external loads.

In some structural applications, FG-GOP reinforced composite beams are subjected to static mechanical loads, hence, it is necessary to perform linear static analysis to examine the behavior of FG-GOP reinforced composite beams under static mechanical loads. The first step to analyze the behavior of composite beams is to use a beam model for theoretical modeling. As an extension of the Euler-Bernoulli beam theory (EBT), the Timoshenko beam model (TBT) [28] called also first-order shear deformation theory (FSDT) is proposed to model beam structures made of composite materials. FSDT takes into account the shear deformation effect and can be applied for modeling both thin and thick composite beams. A high order shear deformation beam theory was utilised by Zghal et al. [29] to study the bending response of FG beams. Frikha et al. [30] have proposed a new high order shear deformation beam model to analyze the behavior of FGM beams under static loads. The vibrational response of FG beams was studied using a mixed formulation [31]. A simple FSDT-based meshfree method was developed to perform static bending and free vibration investigation of functionally graded plates [32]. A new simple FSDT theory in conjunction with the level set method is adopted to examine the buckling and free vibration problems of laminated composites plates [33]. Zhang et al. [34] have investigated the bending, buckling, and free vibration behaviors of FG-GOP reinforced composite beams using FSDT and an analytical approach. In [35], the nonlinear static bending of functionally graded polymer-based circular microarches reinforced with graphene oxide nanofillers was analyzed using an exponential shear deformation theory in conjunction with the Ritz method. A new Ritz-solution shape function was proposed to study the vibration behavior of functionally graded graphene oxide-reinforced composite beams [36]. Ebrahimi et al. [37] have studied the vibration response of nanocomposite beams strengthened with graphene oxide powder subjected to a magnetic field using an analytical solution based on Galerkin's method. The free vibration behavior of porous nanocomposite beams reinforced with multi-scale hybrid graphene oxide powder and carbon fiber was examined using the finite element method [38]. The post-buckling response of a concrete beam reinforced with graphene oxide powders and resting on an elastic medium was analyzed taking into account geometrical imperfections [39]. A higher-order refined beam model was adopted to study the linear buckling behavior of FG-GOP nanocomposite beams [40]. The dynamic stability response of FG polymer microbeams reinforced with GOP was studied using Chebyshev polynomial-based Ritz method [41]. The thermal buckling response of porous functionally graded composite beams reinforced with graphene platelets is investigated using the generalized differential quadrature method [42]. A layer-wise formulation model was used to analyze the vibrational behavior of multilayer functionally graded polymer composite plates reinforced with carbon-based nanofillers [43]. The wave propagation in functionally graded porous composite plates reinforced with graphene platelets was examined [44].

Generally, the access to static mechanical behaviors of FG-GOP reinforced composite beams can be mainly realized using analytical or numerical approaches. However, it is not easy to obtain analytical solutions for composite structures with complex geometries and various boundary conditions and loads. Thus, numerical techniques have been proposed as an efficient alternative approach to find approximate solutions. Besides the classical numerical procedures which are named mesh-based methods (MBMs) such as the well-known finite element method (FEM) [45], new advanced discrete numerical techniques called mesh-free or meshless methods (MFMs) [46] have appeared these last years. In MFMs, the problem domain and its boundaries are represented by a set of nodes uniformly or nonuniformly distributed, these nodes are not linked to each other in a predefined fashion (mesh), and, thus, no nodal connectivity between them is required. This characteristic offers MFMs several advantages such as the treatment of nonlinear deformation analysis of structures and modeling domains with complex geometries. In MFMs, two procedures called collocation and Galerkin can be used to obtain the global discrete system of equations. In meshfree Galerkin methods or MFMs based on weak form, the background mesh is necessary to integrate the weak form of equations, which are costly, from a computational point of view, and impose some difficulties. To avoid the use of background mesh and numerical integration, meshfree collocation methods or MFMs based on the strong form are most preferred. In meshfree collocation methods, the strong form of the governing equations is approximated without the use of weak formulation and, hence, no numerical integration is needed. These methods can be considered as truly MFMs. Several authors have adopted MFMs to examine the behavior of composite materials. Bui et al. [47] have used the radial point interpolation method to explore the transient and natural frequency responses of sandwich beams with an inhomogeneous functionally graded core. The moving Kriging interpolation method is adopted to study the vibration behavior of laminated composite plates [48].

Among MFMs collocation methods, the multiquadric radial basis function (MQRBF) method [49] is the most efficient and used one. The MQRBF was originally introduced by Hardy for data surface fitting [50] and then adopted by Kansa for solving partial differential equations [51]. The MQRBF was extended by Ferreira for the analysis of composite beams, plates and shells [52], [53], [54]. MQRBF has several interesting advantages such as the absence of nodal connectivity, its mathematical simplicity, ease of implementation, and use. Additionally, as the MQRBF belongs to the MFMs collocation methods, there is no need for numerical integrations, which reduces the computational cost compared to MFMs based on weak formulation. In comparison with other existing meshfree methods, the computation of derivatives for the MQRBF shape function is simple and straightforward. Moreover, MQRBF shape functions verify the Kronecker delta property and are with infinite continuity, hence, the numerical model developed based on MQRBF is free of shear locking phenomena and the imposition of boundary conditions is straightforward. The MQRBF is easily adaptable to irregular domains or higher dimensions. Nevertheless, MQRBF shape functions suffer from some limitations. The value associated with the MQRBF shape parameter affects significantly the accuracy and stability of approximation solutions, and, consequently, should be chosen carefully. Also, MQRBF interpolation matrix is usually full and badly ill-conditioned especially when the number of nodes becomes large. In earlier works, the authors have applied successfully the MQRBF collocation technique combined with the asymptotic numerical method (ANM) to study the nonlinear bending of single-walled carbon nanotubes [55] and FG porous beams [56].

A literature survey shows that there is a lack of studies on the linear bending response of functionally graded graphene oxide powder-reinforced composite beams. Furthermore, we have found that the globally supported MQRBF approximated technique has been used often and successfully in the analysis of FG structures while its use for structural investigation of functionally graded materials reinforced with CNTs, GPLs, and GOPs is limited. Thus, the novelty of the present paper is to use a meshfree collocation method based on the MQRBF approximation technique to study the bending behavior of FG-GOP reinforced composite beams subjected to two different static mechanical loads and various boundary conditions. It should be noted that the present work is partially similar to the work in reference [34], but in contrast to this interesting work where the authors have used an analytical approach to study the behavior of FG-GOP reinforced composite beams subjected to uniformly distributed transverse load with only two types of boundary conditions. Here, we use a numerical procedure based on the MQRBF approximation technique to investigate the behavior of FG-GOP reinforced composite beams under uniform and non-uniform distributed loads with various boundary conditions. In this paper, the effective Young's modulus of composite beams is calculated using a modified Halpin-Tsai model and the effective Poisson's ratio is determined using the rule of mixture. To demonstrate the robustness and accuracy of the proposed numerical model based on MQRBF, various comparative studies are provided. Some numerical tests are presented to examine the effect of total weight fraction, distributions and dimensions of GOP reinforcement, loads, boundary conditions and slenderness ratio on the bending behavior of FG-GOP reinforced composite beams.

This paper is organized as follows: The theoretical formulation of FG-GOP reinforced composite beams using FSDT is presented in section 2. The numerical solution strategy based on MQRBF is detailed in section 3. Validation of the used numerical model and parametric studies are given in section 4. This work ends with the conclusion that is highlighted in section 5.

Section snippets

Theoretical formulation

This section is reserved for the theoretical formulation of the first-order shear deformation theory FSDT of beams adopted in this paper for modeling functionally graded composite beams structures reinforced with graphene oxide powder.

Discretization of equilibrium equations and boundary conditions using MQRBF

In this work, and in order to calculate the generalized displacements and stresses of the FG-GOP reinforced composite beams subjected to mechanical loads and various boundary conditions, the matrix strong form system of equations (Eq. (31)) governing the behavior of composite beams (including boundary conditions, constitutive equations, and equilibrium equations) are solved numerically using a meshless collocation technique based on the globally supported MQRBF approximation method which is one

Numerical results and discussion

This section is devoted to analyze the linear bending response of straight, thin and moderately thick functionally graded FG composite beams reinforced with graphene oxide powder GOP subjected to external static mechanical loads with different boundary conditions. The results obtained with the proposed numerical approach based on MQRBF are compared with those of the literature [34], [59], [60], [61] computed numerically and analytically. In addition, the effects of length-to-thickness ratio,

Conclusion

In this paper, the bending behavior of straight, functionally graded graphene oxide powder-reinforced composite beams subjected to two types of static mechanical loads, and different boundary conditions is investigated using a meshfree collocation method based on the MQRBF approximation technique and the first-order shear deformation beam theory. The effective Young's modulus is estimated using a modified Halpin-Tsai model and the effective Poisson's ratio is determined by the rule of mixture.

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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