Nonlinear bending analysis of arbitrary-shaped porous nanocomposite plates using a novel numerical approach

doi:10.1016/j.ijnonlinmec.2020.103556

International Journal of Non-Linear Mechanics

Volume 126, November 2020, 103556

https://doi.org/10.1016/j.ijnonlinmec.2020.103556 Get rights and content

Highlights

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Nonlinear bending analysis of arbitrary-shaped FG-GPLRC porous plates with cutout.
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Derivation of governing equations using HSDT in a new vector-matrix form.
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Employing VDQFEM for problems with variously-shaped polygon and concave domains.
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Using mixed method to guarantee continuity of derivatives on boundaries of elements.
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Studying effects of porosity and GPL distributions and porosity coefficient.

Abstract

In this paper, an efficient numerical strategy is used to study the geometrically nonlinear static bending of functionally graded graphene platelet-reinforced composite (FG-GPLRC) porous plates with arbitrary shape. Porous nanocomposite plates including cutout with various shapes can be modeled by the present approach. Four types of porous distribution scheme and four GPL dispersion patterns are selected, and the material properties are calculated based on the closed-cell Gaussian random field scheme, the Halpin–Tsai micromechanical model together with the rule of mixture. First, the variational statement of governing equations based on the virtual work principle and higher-order shear deformation theory (HSDT) is derived and presented in vector–matrix form for computational aims. Then, using the ideas of variational differential quadrature and finite element methods (VDQ and FEM), a numerical approach called as VDQ-FEM is used to address the considered problem. In VDQ-FEM, the domain of problem is first transformed into a number of finite elements. In the next step, the VDQ discretization technique is implemented within each element. Then, the assemblage procedure is performed to obtain the set of Studying effects of porosity and GPL distributions and porosity coefficient matricized governing equations which is finally solved by means of the pseudo arc-length continuation algorithm. One of the main novelties of the present work in implementing VDQ-FEM is proposing an efficient way based on mixed-formulation to guarantee the continuity condition of first-order derivatives in entire domain for the used HSDT model. A detailed parametric study is conducted to investigate the nonlinear bending of FG-GPLRC porous plates with different shapes. In the numerical results, the effects of porosity coefficient, porosity distribution pattern, GPL distribution pattern and boundary conditions on the nonlinear bending response of plates are analyzed.

Introduction

Plates are important engineering elements that are widely used in various industrial applications. Hence, investigation into the mechanical behaviors of such structures under different kinds of loading conditions is of great importance from the design standpoint. Moreover, the recent developments of nanotechnology have led to the production of engineering structures made of nanocomposites with excellent mechanical properties. For example, it was shown that adding a small amount of carbon nanotubes (CNTs) can significantly enhance the thermo-electro-mechanical properties of polymer nanocomposites [1], [2], [3], [4]. Several research works have been also conducted on the mechanical performance of polymer composites reinforced by graphene or its derivatives [5], [6], [7], [8]. For instance, Liang et al. [7] revealed that addition of graphene oxide with 7% weight fraction can considerably improve the elastic modulus and tensile strength of graphene oxide-reinforced polymer nanocomposite. Besides, using functionally graded CNT and graphene platelets reinforced composites (FG-CNTRCs & FG-GPLRCs) has attracted a lot of attention in this area. Therefore, a wide range of research works has been carried out on the static and dynamic structural analysis of FG-CNTRC [9], [10], [11], [12], [13], [14], [15] and FG-GPLRC [16], [17], [18], [19] beams, plates and shells. In the following, some of the recent papers are reviewed.

Nguyen et al. [9] studied the bending and free vibration behaviors of FG-CNTRC shells based upon the first-order shear deformation theory (FSDT). They proposed the use of non-uniform rational B-Spline (NURBS) basis functions to address the mentioned problems. Civalek and Baltac $ı$ oğlu [10] adopted the discrete singular convolution method in the vibration analysis of FG-CNTRC annular sector plates under various boundary conditions. Wu et al. [14] analyzed nonlinear primary and super-harmonic resonances of FG-CNTRC beams within the framework of Timoshenko beam theory using the Galerkin and incremental harmonic balance methods. Through developing the weak form of nonlinear governing equations and using a solution approach based on the differential quadrature technique, Gholami and Ansari [16] studied the geometrically nonlinear forced vibration of rectangular plates made of FG-GPLRCs. They investigated the effects of GPL reinforcement and geometrical parameters on the frequency– and force–response curves of FG-GPLRC plates with different boundary conditions subjected to harmonic excitation. Based on a semi-analytical approach, Wang et al. [18] analyzed the vibrations of beams made of FG-GPLRC subjected to two successive moving masses based on the higher-order shear deformation theory (HSDT). Zhou et al. [19] developed a higher-order shear deformable shell model to study the nonlinear buckling of FG porous GPLRC cylindrical shells. They obtained explicit expressions of buckling equations for shells under clamped and simply-supported boundary conditions using the Galerkin technique.

In addition, due to their interesting properties including being lightweight and high strength, FG porous composite materials are gaining importance in different fields. Accordingly, several researches on the behaviors of beams [20], [21], [22], [23], [24], plates [25], [26], [27] and shells [28] made of FG porous composites subjected to different environments can be found in the literature. For instance, using the Timoshenko beam theory and the Ritz solution method, Kitipornchai et al. [29] examined the free vibration and buckling of FG porous nanocomposite beams reinforced with GPLs. Also, Chen et al. [30] investigated the nonlinear free vibration and postbuckling of porous FG-GPLRC beams with different boundary conditions.

Recently, a DQ-based numerical approach called as variational differential quadrature (VDQ) has been developed in [31] through which the energy functional in structural mechanics can be directly discretized using matrix differential and integral operators. No needing shape functions to describe the unknown fields, simple implementation, being locking-free, compact matrix formulation, high accuracy and fast convergence rate are the main advantages of VDQ. However, the main limitation of this method is its inability to study polygon domains (e.g. trapezoid) and concave domains (e.g. with holes). Later, the VDQ method was enabled to consider arbitrary-shaped polygon domains using the mapping technique in [32], [33]. Also, in [34], a VDQ-based approach called VDQ-finite element method (FEM) was proposed for the nonlinear analysis of hyperelastic micromorphic continua with concave domains.

A literature review reveals that investigation on the linear and nonlinear mechanical behaviors of nanocomposite plates reinforced with graphene nanoplatelets are limited to those with basic shapes such as rectangular, circular and annular ones; and fewer attentions are devoted to the nanocomposite plates with complex geometries. To authors’ best knowledge, the nonlinear bending of FG-GPLRC porous plates with different geometries has not been studied in the literature up to now. Therefore, the main aim of the present study is to address the nonlinear bending of plate-type structures made of FG-GPLR porous nanocomposites in the context of HSDT via VDQ-FEM. Plates with arbitrary shapes including hole can be modeled using the proposed approach. Various kinds of porous distribution scheme and GPL dispersion patterns are considered. To calculate the effective material properties of nanocomposite, the closed-cell Gaussian Random field scheme, the Halpin–Tsai micromechanical model accompanied by the rule of mixture are utilized. To implement VDQ-FEM, the variational form of governing equations is obtained first. The relations are presented in a novel vector–matrix form for computational purposes. The domain of problem is then transformed into a number of finite elements. In the next step, the VDQ discretization technique is implemented within each element. Then, the assemblage procedure is performed to derive the set of matricized governing equations that is finally solved by means of the pseudo arc-length continuation algorithm. Selected numerical results are presented for the nonlinear bending of FG-GPLR porous nanocomposite plates with various shapes and boundary conditions. The influences of porosity coefficient, porosity distribution pattern, GPL distribution pattern and boundary conditions are investigated in the numerical examples.

It should be noted that based on the used HSDT model, the second-order derivatives are appeared in the governing equations which necessitates the continuity of first-order derivatives on the common boundaries of elements. The Hermite shape functions are often used in FEM to ensure this continuity. Since it is not possible to define shape functions for unknown fields in the VDQ technique, the mixed formulation approach is used herein to accommodate the continuity to be imposed. Based upon this approach, the order of existing derivatives is reduced by defining the first-order derivatives of related field as new degrees-of-freedom and also their initialization in entire domain using the Lagrange constraint approach. Accordingly, the necessity for continuity of first-order derivatives is bypassed through an effective manner.

Section snippets

Material properties

The material properties of porous nanocomposite plate change in the thickness direction because of variation of density and size of internal pores in the manufacturing process [26], [29], [30]. Four kinds of porous distribution scheme are taken into account. Furthermore, four GPL dispersion patterns are selected for each porosity distribution. The variation of elastic modulus $E (X_{3})$ , shear modulus $G (X_{3})$ and mass density $ρ (X_{3})$ along the thickness direction for the considered porosity distributions are

Vector–matrix and discretized form

The tensor relations are expressed herein in compact vector–matrix form as it can be efficiently used in numerical approaches. The vector–matrix relations are then discretized on space domain. The discretization process is performed by means of the operator ${[[■_{1}]]}_{■_{2}}$ which discretizes the parameter $■_{1}$ on the space $■_{2}$ . The concepts of discretization can be found in [31]. The following spaces and parameters are also considered for discretization of relations: $\begin{matrix} \{\begin{matrix} A - space (thickness direction) : & in X_{3} direction \\ B - \end{matrix} \end{matrix}$

Strain energy term

Tensorial form of potential energy and variation of potential energy are $w^{e l a s} = \frac{1}{2} σ : ε,$ $Π^{e l a s} = \int_{ℭ} w^{e l a s} (ε) d \forall,$ $δ Π^{e l a s} = \int_{ℭ} δ w^{e l a s} d \forall = \int_{ℭ} (\frac{\partial w^{e l a s}}{\partial ε} : δ ε) d \forall = \int_{ℭ} (σ : δ ε) d \forall$ whose vector–matrix and discretized form can be derived as $δ Π^{e l a s} = \int_{ℭ} (\hat{σ} \cdot δ \hat{ε}) d \forall = \int_{ℭ} (δ d^{T} E^{l^{T}} P^{T} \hat{C} P E^{l} d + δ d^{T} E_{1}^{{n l}^{T}} C_{1} P_{1} E_{1}^{n l} d + \frac{1}{2} δ d^{T} E_{1}^{l^{T}} P_{1}^{T} C_{1} E_{1}^{n l} d + \frac{1}{2} δ d^{T} E_{1}^{{n l}^{T}} C_{1} E_{1}^{n l} d) d \forall = \int_{B} (δ d^{T} E^{l^{T}} C_{1}^{e q} E^{l} d + δ d^{T} E_{1}^{{n l}^{T}} C_{2}^{e q} E_{1}^{n l} d + \frac{1}{2} δ d^{T} E_{1}^{l^{T}} C_{3}^{e q} E_{1}^{n l} d + \frac{1}{2} δ d^{T} E_{1}^{{n l}^{T}} C_{4}^{e q} E_{1}^{n l} d) d A = {\overset{2 d}{S}}_{X_{1} X_{2}} [[δ d^{T} (E^{l^{T}} C_{1}^{e q} E^{l} + E_{1}^{{n l}^{T}} C_{2}^{e q} E_{1}^{n l} + \frac{1}{2} E_{1}^{l^{T}} C_{3}^{e q} E_{1}^{n l} {+ + \frac{1}{2} E_{1}^{{n l}^{T}} C_{4}^{e q} E_{1}^{n l}) d]]}_{B} = δ d^{T} (K_{d d}^{l} + K_{d d}^{n l}) d = δ d^{T} K_{d d}^{e l a s} d$ where $C_{1}^{e q} = \int_{A} P^{T} \hat{C} P d X_{3} = P^{T} (I_{3} ⊛ 〈\overset{1}{S}〉)$

Virtual work principle

The virtual work principle states that $Π = Π^{e l a s} + Π^{l a g} - Π^{e x t},$ $δ Π = δ Π^{e l a s} + δ Π^{l a g} + δ Π^{e x t} = 0,$ $δ Π = δ d^{T} K_{d d}^{e l a s} d + δ d^{T} K_{d λ}^{l a g} \overset{=}{λ} + δ {\overset{=}{λ}}^{T} K_{λ d}^{l a g} d - δ d^{T} F_{d}^{e x t} = δ d^{T} (K_{d d}^{e l a s} d + K_{d λ}^{l a g} \overset{=}{λ} - F_{d}^{e x t}) + δ {\overset{=}{λ}}^{T} K_{λ d}^{l a g} d = 0 .$ Using the fundamental lemma of calculus of variation leads to $\{\begin{matrix} K_{d d}^{e l a s} d + K_{d λ}^{l a g} - F_{d}^{e x t} = 0 \\ K_{λ d}^{l a g} d = 0, \end{matrix}$ $[\begin{matrix} K_{d d}^{e l a s} d + K_{d λ}^{l a g} - F_{d}^{e x t} \\ K_{λ d}^{l a g} d \end{matrix}] = [\begin{matrix} K_{d d}^{e l a s} & K_{d λ}^{l a g} \\ K_{λ d}^{l a g} & 0 \end{matrix}] [\begin{matrix} d \\ \overset{=}{λ} \end{matrix}] - [\begin{matrix} F_{d} \\ 0 \end{matrix}] = K^{*} d^{*} - F^{*} = 0$ $R = K^{*} d^{*} - F^{*} = 0$ To solve the nonlinear bending problem, it is assumed that transverse static load is uniformly distributed through the thickness of plate.

Results and discussion

In this section, a detailed parametric study is conducted to reveal the nonlinear bending of GPL-reinforced porous metal-matrix nanocomposite plates with the material properties and GPL dimensions as $E_{M} = 130 GPa$ , $ρ_{M} = 8960 kg ∕ m^{3}$ , $ν_{M} = 0.34$ for copper [36], [37], and $E_{G P L} = 1.01 TPa$ , $ρ_{GPL} = 1062.51 kg ∕ m^{3}$ , $ν_{G P L} = 0.186$ , $w_{G P L} = 1.5 μ m$ , $l_{G P L} = 2.5 μ m$ , and $t_{G P L} = 1.5$ nm for GPLs [38], [39].

Also, various boundary conditions including SSSS, CCCC, CSCS, CCSS and SCSC are considered to generate the numerical results.

First,

Conclusion

In this article, in the context of Reddy’s third-order shear deformation theory, a numerical variational approach named as VDQ-FEM was utilized to analyze the geometrically nonlinear bending response of plate-type structures made of FG-GPLRC porous nanocomposites. Various patterns were assumed for the distribution of porosity and GPL nanofillers. The presented approach, which simultaneously uses the advantages of VDQ and FEM, can be utilized for plates with arbitrary shapes including cutout.

CRediT authorship contribution statement

R. Ansari: Supervision, Conceptualization, Writing - review & editing. R. Hassani: Conceptualization, Methodology, Validation. R. Gholami: Methodology, Validation. H. Rouhi: Methodology, Writing - original draft.

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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Nonlinear bending analysis of arbitrary-shaped porous nanocomposite plates using a novel numerical approach

Highlights

Abstract

Introduction

Section snippets

Material properties

Vector–matrix and discretized form

Strain energy term

Virtual work principle

Results and discussion

Conclusion

CRediT authorship contribution statement

Declaration of Competing Interest

Comput. Mater. Sci.

Composites A

Composites B

Composites B

Compos. Struct.

Compos. Struct.

Comput. Methods Appl. Mech. Engrg.

Composites B

Aerosp. Sci. Technol.

Int. J. Mech. Sci.

Eng. Struct.

Appl. Math. Model.

Compos. Struct.

Int. J. Mech. Sci.

Compos. Struct.

Acta Astronaut.

Compos. Struct.

Compos. Struct.

Aerosp. Sci. Technol.

Compos. Struct.