Large amplitude free flexural vibrations of functionally graded graphene platelets reinforced porous composite curved beams using finite element based on trigonometric shear deformation theory

Highlights

• Provided the frequency-amplitude relation for porous GPL reinforced curved beams.

• Shown the effect of distribution of pores/GPLs on the nonlinear frequency.

• Presented results varying slenderness ratio and different boundary conditions.

• Observed change in degree of hardening trend due to the inclusion of porosity/GPL.

• Included new results for benchmarking the solutions from other numerical approach.

Abstract

In this paper, the large amplitude free flexural vibration characteristics of fairly thick and thin functionally graded graphene platelets reinforced porous curved composite beams are investigated using finite element approach. The formulation includes the influence of shear deformation which is represented through trigonometric function and it accounts for in-plane and rotary inertia effects. The geometric non-linearity introducing von Karman’s assumptions is considered. The non-linear governing equations obtained based on Lagrange’s equations of motion are solved employing the direct iteration technique. The variation of non-linear frequency with amplitudes is brought out considering different parameters such as slenderness ratio of the beam, curved beam included angle, distribution pattern of porosity and graphene platelets, graphene platelet geometry and boundary conditions. The present study reveals the redistribution of vibrating mode shape at certain amplitude of vibration depending on geometric and material parameters of the curved composite beam. Also, the degree of hardening behaviour increases with the weight fraction and aspect ratio of graphene platelet. The rate of change of nonlinear behaviour depends on the level of amplitude of vibration, shallowness and slenderness ratio of the curved beam.

Introduction

The use of advanced materials is necessitated in aerospace technology due to the requirements of high stiffness-to and strength-to-weight ratios. Furthermore, the progress in material science along with the processing technology has led to the increased usage of porous metals and metallic foams for various applications [1], [2], [3], [4]. The properties of such materials can be significantly improved with the addition of nano-fillers in the form of carbon nanotubes (CNTs) and graphene platelets (GPLs). The inclusion of nano-fillers effects the change in the global properties of these metallic structures depending on the weight fraction and distribution of fillers [5], [6], [7]. The structural members like beams, plates and shells made of reinforced porous materials or foams can form the integral part of various structural systems in aerospace engineering. These structural members, when subjected to supersonic flow, can experience large amplitude of vibration leading to catastrophic failure. Hence, the investigation of such porous structural members under dynamic situation continues to attract researchers for the design of reliable aerospace structural systems. Since the proposed study is related to the curved beams, the brief discussion on available studies in the literature on the large amplitudes of vibration is here restricted to beams.

The nonlinear vibration of isotropic straight beams has been studied in the past [8], [9], [10], [11], [12], [13], [14], [15] based on various formulations such as Lagrangian-type [8] and Ritz formulation [9]. Ref. [11] brought out the approximation and assumption involved in various formulations and solution methodologies. Rao et al. [13] studied the non-linear vibration of beams using an eigenvalue approach by reducing the non-linear strain–displacement relations to an appropriate linear algebraic problem. Patel et al. [14] investigated the nonlinear free vibration of cantilever beams and brought out the in-plane inertia effect on the nonlinear behaviour and the change in mode shape with amplitudes. Pagani et al. [15] recently highlighted the change in nonlinear behaviour and mode change in the large deflection and post-buckling of thin-walled beams. The increased use of advanced composite materials has also recently motivated the researchers to investigate the nonlinear dynamic behaviour of laminated composite structures. Kapania and Raciti [16] analysed the free vibration of unsymmetrical laminated composite beams while Iu et al. [17] carried out the nonlinear forced vibration of beams taking into consideration of damping effects. Singh et al. [18] extended the scope of the work presented in [16] considering higher-order theories. Here, the governing equations were obtained assuming the mode shapes and solved utilizing a direct numerical integration approach. Bassiouni et al. [19] examined the dynamic response of laminated composite beams through the experimental study and theoretical approach ensuring the displacement compatibility between laminas. A detailed review of available work on the vibration of composite beams was documented by Hajianmaleki and Qatu [20]. The investigation concerning dynamic characteristics of functionally graded material (FGM) beams has been carried out with great interest in recent years. Ke et al. [21] analysed the large amplitude vibration behaviour of Euler–Bernoulli laminated composite beams employing the Galerkin procedure and a direct numerical integration scheme coupled with the Runge–Kutta iterative method. The effect of material distributions on such behaviour was also highlighted. Kitipornchai et al. [22] presented a parametric study on the non-linear dynamic behaviour of cracked FGM Timoshenko beams accounting transverse shear and inertia effects. The governing equations were derived using the Ritz method in conjunction with the direct iterative method in obtaining the non-linear frequencies. Wu et al. [23] described the non-linear vibration of functionally graded carbon nanotube (FG–CNT) reinforced beams and their sensitivity analysis considering geometric imperfections based on the formulation outlined in [22]. Feng et al. [24] attempted the large amplitude vibration behaviour of Timoshenko FGM beams reinforced with graphene platelets. The governing differential equations, obtained using Hamilton’s principle, were solved analytically employing the Ritz method coupled with the direct iteration technique. It may be noted in this discussion that all these studies are concerned with the straight beam analysis.

Few notable contributions to the study of isotropic curved beams or shallow arches in the past discussed in Refs. [25], [26], [27], [28]. In Ref. [25], a modified modal equation for curved beams based on the Duffing equation was formulated and the solution was obtained by integration using a fourth order Runge–Kutta method. A mixed type finite element approach using Reissner-type variational statement was proposed in Ref. [26] whereas an analytical formulation introducing Lagrangian variational method was employed in [27]. A shear flexible field consistent curve beam element based on a spline function was examined for the nonlinear dynamic analysis of curved beams in [28]. Ganapathi et al. [29] studied the large amplitude vibrations of laminated composite curved beams extending the element proposed in [28] and proposing the Newton–Raphson iterative method to achieve the required convergence. They concluded that the type of nonlinearity can alter with respect to amplitude of vibration and geometric parameter of curved beam. At higher amplitude of vibration, the occurrence of redistribution of mode shape and the possibility of higher modes are reported. Hajianmaleki and Qatu [30] developed a general approach for studying the vibration of curved beams by implementing curvature terms in the strain–displacement equations. It may be opined, to the best of authors’ knowledge, there is no study reported in literature on the large amplitude vibrations of functionally graded graphene or CNT reinforced curved beams.

In the present work, the non-linear free flexural vibration of functionally graded graphene platelets reinforced porous composite beams is investigated based on a trigonometric refined model as described by Touratier [31] and Ganapathi et al. [32]. The governing equations developed are solved using a direct iteration method. The formulation developed here is validated against the available work in the literature. A detailed numerical study is carried out considering various parameters such as the effect of slenderness ratio of the beam, curved beam included angle, distribution pattern of porosity and graphene platelets, and boundary conditions on the frequency–amplitude relationship.

The paper is structured as follows: the material aspects of graded porosity distribution and GPL patterns are shown in Section 2, followed by describing the curved beam theory and governing equations in Section 3. The solution methodology is briefly highlighted in Section 4. Section 5 comprises the assessment of the formulation and the detailed numerical studies followed by the conclusion.