Nonlinear forced vibration analysis of laminated composite doubly-curved shells enriched by nanocomposites incorporating foundation and thermal effects
Introduction
A key issue in several engineering aspects such as aeronautics, aerospace and shipping industries is evaluation of dynamic behaviour of thin-walled structures when they are subjected to dynamic external forces incorporating the geometrical nonlinearity effects. Among the various types of thin-walled members, doubly-curved shallow shell panels are mostly used in the recent decays which their large amplitude vibration has been investigated in several studies. The main concepts of geometrical nonlinear vibration of doubly-curved shallow shells have been basically explained in Refs. [1], [2]. However, in order to extend the fundamental equations to investigate the large amplitude vibration of such shells, some initial assumptions are required. The first assumption is the basic theory which is related to the proportional thickness of the shells. For instance, Foroutan et al. [3] developed the fundamental differential equations of the carbon nanotube (CNT) reinforced thin cylindrical panels based on the classical theory (CLT) to evaluate their nonlinear dynamic behaviour. Zhang et al. [4] employed first-order shear deformation theory (FSDT) for free vibration analysis of the moderately thick shells.
In order to analyse the thick shells, the higher-order shear deformation theories (HSDT) are required. In this regard, the third-order (TSDT) and three refined variable (TRVSDT) shear deformation theories are the most commonly used theories which were employed in Refs. [5], [6], [7]. Beside the proportional thickness, the scale of analysis is the other important factor to extend the fundamental equations. In order to analysis of plates and shells in micro and nano scale, several methods are applicable such as nonlocal [8], strain gradient [9] and nonlocal strain gradient [10] theories.
Several methods with different analytical and numerical approaches are available in the literature to solve the achieved governing differential equations based on the above theories. The analytical methods give more complicated solutions with several restrictions while they do not need to any computational efforts. The analytical solutions can be obtained based on the various methods such as Rayleigh–Ritz [11], [12] and Jacobi-Ritz [13], [14], [15] methods. The other common method to develop the analytical solutions is Galerkin method which can be directly used to solve the nonlinear differential equations. For instance, Shahmohamadi and Kabir [16] employed this method to evaluate the snap-through behaviour of FGM shallow spherical caps. Similar method was used in Refs. [17], [18] for large amplitude dynamic analysis of CNT-reinforced and auxetic shallow doubly-curved shells, respectively. Some other methods have also been proposed by researchers which are applicable to develop the analytical solutions. For example, an analytical solution based on an asymptotic method was developed in Refs. [19], [20] for nonlinear dynamic analysis of the curved panels. Shahmohammadi et al. [21] developed a closed-form formula based on the modified interactions method to evaluate the snap-through behaviour of CNT reinforced shallow spherical caps. However, the easiest analytical method is Navier method based on the expansion of Fourier series which is applicable only for simply-supported panels [22].
In order to eliminate the restrictions of the analytical solutions, various numerical methods have been developed to analyse the shells and plates. The most commonly used numerical method is finite element method (FEM) which is basis of several commercial softwares [23]. Various versions of FEM was proposed in Refs. [24], [25], [26], [27], [28], [29] for mechanical analysis of the shells. The other popular numerical method is the family of mesh-less method which was employed in Refs. [30], [31]. Isogeometric analysis (IGA) is the other numerical method for analysis of the thin-walled structures with various geometries and boundary conditions [32], [33]. Family of the generalized differential quadrature (GDQ) method is an applicable numerical method employed in Refs. [34], [35].
Using a novel approach, some researchers have developed coupled formulations by combining two numerical methods in order to use their advantages, simultaneously. For example, Shahmohammadi et al. [36], [37], [38], [39] developed a new version of finite strip method (FSM) by combining IGA and FEM in a unite formulation. Mirfatah et al. [40] extended a coupled formulation for analysis of 3D mediums by combining the mesh-free and IGA methods. Rank et al. [41] introduced the finite cell method (FCM) by combining the higher-order versions of FEM and fictitious domain method. Mousavi et al. [42] proposed a coupled IGA-Mesh-less formulation for analysis of the plates with complicated cutouts.
Investigating the introduced numerical methods shows that they can eliminate the restrictions of the analytical methods, but they lead to increase of the computational time. Therefore, some researchers developed semi-analytical methods. For instance, Senjanović et al. [43] developed a FSM formulation by combining the analytical and FEM methods along the two different directions of the shells surfaces. In Refs. [44], [45], [46], [47], the analysis of various shells was performed by an analytical approach while the mechanical specifications was achieved using a numerical method.
The other important factor which affects the nonlinear dynamic behaviour of the shells is the material constituents of the shells. Considering the previous studies shows that the enriching the polymeric resins by nanocomposites such as carbon nanotubes (CNTs) [48], [49] and graphene nanoplatelets (GNPs) [50], [51] can enhance their material characteristics.
In the current study, the geometrically nonlinear forced vibration analysis of the laminated composite doubly-curved shells enriched by nanocomposites incorporating foundation and thermal effects is performed. The considered shells in this paper are assumed to be thin and moderately thick. Therefore, the governing fundamental equations are extended based on FSDT. Reviewing Refs. [52], [53], [54], [55] shows that the mechanical specs of fibre-reinforced composites can be enhanced by enrichment of their resin by nanocomposites.
Different parts of an aircraft engine and aerospace components can be modelled by shell elements with diverse geometries, including cylindrical, conical, doubly-curved and elliptical. According to the authors' best knowledge, there is no research on the nonlinear forced vibration of the doubly-curved laminated composite shallow shell panels enriched by nanocomposites [56], [57], [58], [59], [60], [61], [62].
In this paper, for the first time, the geometrical nonlinear forced vibration of doubly-curved shell panels made of such a material resting on an elastic foundation is investigated using a closed-form semi-analytical solution. The effects of thermal effects in two situations of uniform temperature raise and thermal gradient are also investigated. The numerical studies indicated that the present method needs no time-consuming computational efforts in order to estimate the nonlinear equilibrium path of the proposed shells that could justify its novelty and advantage.
Section snippets
Hybrid composite doubly-curved shallow shell panels
A doubly-curved shallow shell panel is considered here (Fig. 1). The radii of curvature along x and y directions are and , respectively. According to Fig. 1, the states and represent the geometry of plates and shallow cylindrical panels, respectively. Also, based on Fig. 1, two different conditions of and can be assumed for doubly-curved shell panels. It is assumed that the proposed panels contain three orthotropic layers composed of fibres and
Basic equations
The basic equations corresponding to doubly-curved shallow shell panels based on FSDT are presented here. Generally, the basic equations incorporate constitutive, kinematic, equilibrium, and compatibility equations. Based on FSDT, the stress-strain relationship according to the general form of Hooke's law can be expressed as: in which the subscripts m, b, and s refer to membrane, bending, and transversely shear deformations of the shell. The general
Methodology
This paper aims to obtain a closed-form solution for the nonlinear system of differential equations extended in the previous section. In this regard, the proposed shell panel is assumed to be simply supported which can be described by the following expressions:
By estimating the components of the deformation field using the harmonic functions as expressed in Eq. (25), the above boundary conditions can be satisfied.
Numerical results and discussion
By employing the extended closed-form formula for equation of the dynamic motion, and applying the fourth-order Runge-Kutta method to solve it, the nonlinear forced vibration of the proposed shells is performed here in the framework of a comprehensive parametric study. In order to organize the parametric studies, some definitions are presented here. The curvature of the shells can be expressed by defining a reference curvature radius R. Thus, the curvature radii and can be expressed in
Conclusions
By extending a closed-form formula for the nonlinear dynamic equation of motion of the hybrid composite doubly-curved shell panels, their large amplitude forced vibration was investigated. The governing equations of the proposed semi-analytical solution were developed based on the FSDT principals and they were analytically solved using the main concept of the Galerkin method. The mechanical specs of the hybrid composites was characterized based on the Halpin–Tsai model and calculated by
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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