Elsevier

Thin-Walled Structures

Volume 154, September 2020, 106840
Thin-Walled Structures

Full length article
Extremely large oscillation and nonlinear frequency of a multi-scale hybrid disk resting on nonlinear elastic foundation

https://doi.org/10.1016/j.tws.2020.106840Get rights and content

Highlights

  • Nonlinear frequency and large amplitude vibrations of multi-scale hybrid Nano-composites disk resting on elastic foundation under the temperature gradient effect is firstly investigated.

  • The displacement-strains of nonlinear vibration of the multi-scale laminated disk is obtained via third-order shear deformation (TSDT) theory and Von Karman nonlinear shell theory.

  • The matrix material is reinforced with carbon nanotubes (CNTs) or carbon fibers (CF) at the nano or macro scale, respectively.

  • Some benchmark results related to nonlinear frequency and nonlinear dynamic response of nano-composite disk have been presented under mechanical and thermal effects.

Abstract

This is a fundamental study on the nonlinear vibrations considering large amplitude in multi-sized hybrid Nano-composites (MHC) disk (MHCD) relying on nonlinear elastic media and located in an environment with gradually changed temperature feature. Carbon fibers (CF) or carbon nanotubes (CNTs) in the macro or nano sizes respectively are responsible for reinforcing the matrix. For prediction of the efficiency of the properties MHCD's modified Halpin-Tsai theory has been presented. The strain-displacement relation in multi-sized laminated disk's nonlinear dynamics through applying Von Karman nonlinear shell-theory and using third-order-shear-deformation-theory (TSDT) is determined. The energy methods called Hamilton's principle is applied for deriving the motion equations along with Boundary Conditions (BCs), which has ultimately been solved using the perturbation approach (PA) and generalized differential quadrature method (GDQM). At the final stage, the outcomes illustrate that patterns of FG, fibers' various directions, the WCNT and VF factors, top surface's applied temperature and temperature gradient have considerable impact on the MHCD's nonlinear dynamics. Another important consequence is that, the influences of the θ, WCNT and VF amounts on the disk's nonlinear vibrations may be taken into account at the larger amounts of the high deflection element and the negative axial load's impact on the structure's nonlinear dynamics is more extreme. A more general conclusion of this study is that for designing the MHCD should be more attention to the nonlinearity parameter.

Introduction

As a result of some prominent needs in technology and science for boosting the systems' mechanical response [[1], [2], [3], [4], [5]] and performance [6,7], enhancing material properties [[8], [9], [10]] using MHC has been attracting the numerous researchers' attention to provide a considerable improvements in the practical composite structures in construction applications [[11], [12], [13], [14], [15], [16]]. Hence, for achieving desirable properties thermo-mechanically, carbon along with its extractions are taken into consideration as the most effective to enhance structures of engineering applications. The reinforcements' size choices highly depend on the engineers' aims. Many types of composites have components namely matrix and macro-sized reinforcement including consisted carbon fibers (CF) located in certain orientations to boost the structures' mechanical performance. Chakrapani et al. [17] employed a strategy called multiple sizes to analyze the importance of the fibers' directions and laminates' sequence on the CF reinforced composite beams' forced oscillation response which its viscoelasticity has been analyzed through the Kelvin-Voigt equation. Furthermore, they investigated experimental studies verifying the precision of the determined computational outcomes. Buckling and post-buckling behavior of the straight beam enhanced by fibers and located in an environment with hygrothermal feature using the Reddy's model framework have been scrutinized by Emam and Eltaher [18]. At the same time, enhancing composites using nano-fibers instead of macro-sized fibers reports considerable enhancement in the structures' macro-mechanics. Hence, there are a wide range of scientists focusing mainly on the mechanic's structures used carbon nanotubes as reinforcements (CNTR). A numerical model to investigate buckling of CNTR circular and annular plate based on TSDT. They explained that the crucial force determined by their strategy is less than those earned due to the classical model as an outcome of taking into account terms of shear strain has been reported by Maghamikia and Jam [19]. Stability/instability characteristics of the laminated composite/CNT-reinforced composite structure with and without temperature loading were investigated by Ref. [[20], [21], [22], [23], [24], [25], [26], [27], [28], [29]]. Also, with the aid of some theoretical and experimental methods, Refs [[30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44]] investigated the vibrational characteristics of composite structures under various conditions. In the scope of the linear response of an annular disk, rotation in high-speed and locating in thermal location are taken into consideration as the key presumption in the system's numerical simulating for finding the critical rotational velocity of thermos-whirling circular plates. The considerable influence of damping factors on the FG circular plate's stability and transient forced oscillation regarding viscoelasticity of boundaries is disclosed by Alipour [45]. Based on classic model, a solution method has been developed for investigating FG circular plate's vibrations surrounded by a piezoelectric layer by Ebrahimi and Rastgoo [46]. In their further study, Ebrahimi and Rastgoo [47] analyzed FG annular plate's flexural free vibrations covered by layers of piezoelectricity. Shasha et al. [48] reported a new analytical theory based on surface effects including surface elasticity and Kirchhoff model for obtaining a double-layered micro circular plate's vibrations. The surface properties have been considered in their investigation as the key element of their study's novelty. The outcomes illustrated using modified theory demonstrated that the double-layered micro-sized structure vibrations is fairly larger than the single-layered ones. Ghasemi et al. [49] analyzed dynamic performance of a micro cylindrical shell laminated with fiber and metal. In this study, impact of different types of reinforcement is analyzed to boost the frequency characteristics of the microshell in an effective prcedure. Gholami et al. [50] employed a more applicable gradient elasticity theory with the capability of including higher order parameters and the size effect in the analysis of instability of FG cylindrical microshell. Their results confirmed that the radius to thickness ratio and size dependency have considerable impact on the micro-scaled system's stability. Due to the FSD model, Mohammadimehr et al. [51] analyzed a computational research in the static and dynamic stability behavior of a composite circular plate through using a FE model called GDQM. Moreover, they took into consideration a field of thermo-magnet for defining the laminated structures.

In order to scrutinize nonlinear mechanics of circular disk, Keleshteri et al. [52] studied an analysis regarding FG annular plate's large bending which is enhanced through using CNTs, while it is lied on an elastic foundation. They asserted that in their computational investigation, the von Karman models along with shear deformation are applied to serve more precise outcomes. Moreover, they utilized GDQ and Newton-Raphson strategies to solve their relations. The noticeable outcome could be that the thickness profile and value fraction of CNTs may represent a significant influence on the nonlinear annular disk's vibrations. Ansari and Torabi [53] analyzed FG disk's nonlinear dynamics. The considerable resuls extracted from that study is that they reported a GDQ's modified model to solve the FG disk's motion equation and reported the structure's large amplitude. Ansari et al. [54] reported a computational model to analyze the nonlinear dynamic responses of the compositionally disk which is rested on an elastic media. The composite disk which they modeled is a CNT reinforced FG annular plate. Gholami et al. [55] reported the nonlinear behavior GPL reinforced annular plate under a dynamically load and the structure is covered with the Winkler-Pasternak media. Also, the applied Newton-Raphson and a modified GDQ methods to access the graphene reinforced disk's nonlinear bending are presented in Refs [[56], [57], [58], [59], [60], [61], [62], [63]].

For studying mechanics of structures, there are a wide range of models and theories including a classical model called Love model [64] which can be used in investigating thin shells as well as plates. Then, a vast majority of scientists including Donnell [65] and Sanders [66] merged the configurational supposition with linear elasticity model, while they presented their new models. For instance, Leissa [67] has analyzed these theories' decent collection along with their applications; however, the models presented by Leissa were based on specific suppositions. In the shell's classical theory, it is believed that stresses are unchanged through its thickness. Thereby, classic models are unable to report accurate outcomes for structures with more thicknesses due to the assumption made by Leissa. Moreover, HSDT was introduced for compensating classical model's insufficiency. Surprisingly, in Ref. [68,69] the novel presented fields of displacement analyzed various functions of non-polynomial shear strain shape in displacements of in-plane, and for the case of displacement of out-of-plane, a polynomial one is investigated. Furthermore, Refs. [[70], [71], [72]] reported a broad range of new functions of non-polynomial shear strain geometry, which could be realized to be prominent for solving problems of advanced and classical composites due to its accuracy in bending outcomes. Refs. [[73], [74], [75]] presented bending behavior of composite structures employing a novel non-polynomial HSDTs. In another work, Mantari et al. [76] studied the stretching influence in non-polynomial quqasi-3D HSDTs. This study disclosed real requirement for novel functions of non-polynomial shear strain geometry, while in this paper, a few new samples are prepared for the first time.

As the first research, the nonlinear MHCD's forced and free vibrations, located in the gradually-changed temperature and relying on elastic media are explored. The MHCD's mechanicals are determined upon the Halpin-Tsai theory regarding a model of micro-scaled mechanics. Von-Karman and TSDT nonlinearity models are investigated for extracting the motion equations, while they solved them using PA and GDQ methods. Ultimately, the outcomes demonstrate that patterns of FG, fibers' various direction angles, the WCNT and VF factors, temperature changes and top surface's imposed temperature have significant influence on the MHCD's nonlinear dynamics.

Section snippets

Formulations and theories

In this step we have considered a disk with an external radius of b, an inner radius of a along with a thickness of h, relying on a two-factor elastic media as demonstrated in Fig. 1. Since it is explained in the previous section in details, CF and CNTs may be applied, respectively, as macro-sized and Nano-sized reinforcement.

The strategy of homogenization is presented employing Halpin-Tsai theory and a micro-scaled mechanics, while it is composed of a couple of stages. Firstly, composites’

Process of obtaining the solution

For analysing a cylindrical micro panel's dynamic characteristics an FE model called GDQM has been used. Adding result of the function regarding linearly-weighted feature, in all distinct mesh points approximates the derivatives of nth -order versus its pertained distinct points which should be based on the all domain's length [110]. Thus, the function mentioned may be explained as:rf(x)Rr|x=xp=j=1ngij(r)f(Ri)here, weighting factors in the case of derivatives of the nth-order through the

Results

Since the ultimate aim of this research could be improvements in dynamics of a disk, a famous reinforcement called multi-scale hybrid nano-sized composites is applied, while thermo-mechanical properties of these reinforcements along with the epoxy used in this research are disclose in Table 1 [81].

Conclusions

In this work, nonlinear free and forced vibration characteristics of the MHCD relying on nonlinear elastic medium under gradually-changed temperature are explored using PA and GDQ method and TSD theory. The numerical results revealed that:

  • For all BCs, when the amount of the γelement was positive and negative the dynamic behavior of the system tends to have hardening and softening behaviors, respectively.

  • For S–S and C–F boundary conditions, there was an impressive effect from nonlinearity

CRediT authorship contribution statement

Ali Shariati: Methodology, Writing - original draft, Software. Aria Ghabussi: Methodology, Writing - original draft, Software. Mostafa Habibi: Validation, Writing - review & editing. Hamed Safarpour: Validation. Mehran Safarpour: Resources, Project administration. Abdelouahed Tounsi: Resources, Project administration. Maryam Safa: Writing - review & editing.

Declaration of competing interest

The authors declare that they have no conflict of interest.

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