On the dynamics of nanoshells
Introduction
Aragonite, calcite and vaterite are three polymorphs of calcium carbonate (Moudrakovski, 2013). Aragonite, which evolves spontaneously to calcite, has orthorhombic (orthotropic) structure while calcite, with hexagonal structure, is the most stable polymorph (Dante, 2015). Aragonite have been adopted with success in many biocomposites and biominerals such as bone healing (Perrotti, Iaculli, Fontana, Piattelli & Iezzi, 2017). It has several usages in plastic, paper and automobile industries, too (Ramakrishna, Thenepalli & Ahn, 2017). For example, aragonite whiskers have been adopted as fibers for composites with polymer matrix in industrial usages (Kelly & Zweben, 1968). Naturally, it also exists as the calcareous endoskeleton of corals and in almost all mollusk shells (Moudrakovski, 2013). The most common form of aragonite is like a rod but other aragonite morphologies have been produced, too (Ramakrishna et al., 2017). According to the orthotropic structure of aragonite with considering the shape of mollusk shells and other aragonite structures which have the ability to be fabricated, an orthotropic doubly-curved shell may be chosen for modeling aragonite.
One of the most common structures used extensively in the industry is shell because it is well-known that shells are stiffer and stronger compared to other straight structures. For instance, a cylindrical shell can be found in the body of monocoque in an airplane while spherical or elliptical shells are the desirable geometries for the dishes used in satellite (Bryan, 2018). Roughly speaking, shell may be considered as a mixture of plate and membrane which means that it sustains out-plane loading with both the bending and the plane stresses. One of the types of shells which attracts the attention of some researchers is doubly-curved shell. Doubly-curved shells, for example a part of sphere, have completely different behaviors compared to single curvature shells which are curved on only one linear axis, for example a part of cylinder. Up to now, several different researches have been done on doubly-curved shells.
Tornabene et al. (Tornabene, Fantuzzi, Viola & Reddy, 2014) studied the dynamics of laminated doubly-curved shells and panels adopting a numerical solution procedure using differential quadrature method. In another work, they reported the influence of agglomeration on the vibration of the carbon nanotubes reinforced laminated composite shells (Tornabene, Fantuzzi, Bacciocchi & Viola, 2016). Nguyen and his co-workers (Nguyen, Thai, Luu, Nguyen-Xuan & Lee, 2019) proposed a non-uniform rational B-Spline basis functions and the first-order shear deformation shell theory to obtain the nonlinear equation system in studying a reinforced composite shell. Wang et al. (Wang, Shi, Liang & Pang, 2017) utilized a higher-order shell model to analyze the free vibration of doubly-curved panel which solved by the Fourier–Ritz method. Dynamics of laminated doubly-curved shell including general boundary conditions were reported by Guo et al. (Guo, Shi, Wang, Tang & Shuai, 2018). There are also a few studies which have been done on anisotropic doubly-curved shell panels. Dynamics of anisotropic doubly-curved shells of revolution defined by free-form meridians were studied by Tornabene et al. (Tornabene, Liverani & Caligiana, 2011, 2012).
In the last two decades, nanoscience has attracted the attention of many industry researchers. The reason for this is the unique properties of nanoscale materials in comparison with macro ones. Industry support for nanoscience researches and, of course, researchers' personal interest has motivated numerous studies in this field. These researches have dealt with this issue from different aspects. For examples, some works have paid attention to the production of nanostructures (Hinman & Suslick, 2017), some others have focused on the study of their properties (Ramirez et al., 2016, Chen et al., 2018, Yu et al., 2020 Yan et al., 2020), some of those have investigated the methods of modeling them (Zehetbauer & Estrin, 2009), and many others have paid attention to other aspects of the nanostructure (Guo et al., 2020; Guo, Qian, Cai, Tang & Liu, 2019; Liu et al., 2020; Luo et al., 2020; Wang et al., 2018). Modeling performed on nanostructures has been done with different approaches such as molecular dynamics (Aydogdu, 2012; Cao, Chen & Guo, 2006; Qiu, Bao, Zhang, Wu & Ruan, 2012; Yang, Kim & Cho, 2018), molecular structural mechanics (Eberhardt & Wallmersperger, 2019; Firouz-Abadi, Moshrefzadeh-Sany, Mohammadkhani & Sarmadi, 2016; Hou, Deng & Zhang, 2017; Kazemi, Hajiahmadi & Rajabi, 2019; Korobeynikov, Alyokhin & Babichev, 2018; Narendar, Mahapatra & Gopalakrishnan, 2011; Wang, Xia & Tan, 2016) and size-dependent continuum theories (Barretta & de Sciarra, 2019; Barretta, Faghidian & de Sciarra, 2019; Darban, Luciano, Caporale & Fabbrocino, 2020; Eyvazian, Shahsavari & Karami, 2020; Faghidian, 2018; Farajpour, Farokhi, Ghayesh & Hussain, 2018, 2019; Ghayesh, 2019; Ghayesh & Farokhi, 2018, 2019; Ghayesh, Farokhi & Farajpour, 2019, 2018; Gholipour & Ghayesh, 2020; Karami & Janghorban, 2019; B. Karami, Janghorban & Tounsi, 2018; Karami, Shahsavari, Janghorban & Li, 2019; Khaniki, 2019; Li & Hu, 2016, 2015; Li, Lin & Ng, 2020, 2018; Malikan & Eremeyev, 2020; Malikan, Krasheninnikov & Eremeyev, 2020; Pinnola, Faghidian, Barretta & de Sciarra, 2020; Shafiei, Mousavi & Ghadiri, 2016; Shahverdi & Barati, 2017; X. Zhu & Li, 2017, 2017). The size-dependent continuum theories have attracted the most attention among researchers not because it's the most accurate method, but because of its simplicity, availability and familiarity of researchers with them. Achilles heel of this method is that the parameters used in these theories which should be adopted to capture size effects, which is necessary in modeling many nano-sized structures, are unknown up to now. Because of this problem, these theories are difficult to use for industrial usage. Solving this problem is a big step in industrializing this approach. The following is a summary of some of the work done considering this approach for static and dynamics of nanoshell.
The resonance phenomenon of a graphene nanoplatelets reinforced composite doubly-curved nanoshell were proposed by Karami et al. (Karami & Shahsavari, 2020). They have also reported the behavior of vibration and elastic bulk waves of porous doubly-curved nanoshell (Karami, Shahsavari & Janghorban, 2019, 2019). Free vibration of a reinforced composite doubly-curved nanoshell were investigated by Dindarloo and Li (Dindarloo & Li, 2019). Karami and his co-authors studied the elastic bulk waves, static bending and buckling in anisotropic and functionally graded anisotropic doubly-curved nano-size shell panels (Karami & Janghorban, 2020; B. Karami, Janghorban & Tounsi, 2018; Karami, Janghorban & Tounsi, 2020, 2019). A nonlocal model was proposed to study the free vibration of doubly curved piezoelectric nanoshell by Arefi (Arefi, 2018). Malikan and his co-workers (Malikan, Uglov & Eremeyev, 2020) proposed a non-classical model based on nonlocal strain gradient theory to study the instabilities and post-buckling of piezomagnetic and flexomagnetic nanostructures. Zhu et al. (Zhu, Fang & Yang, 2019) investigated the free vibration of functionally graded viscoelastic piezoelectric doubly curved nano-size shell including nonlinear terms in strains as well as surface effects. They have also proposed a new methodology to control the size-dependent nonlinear vibration smartly in a viscoelastic orthotropic piezoelectric doubly-curved nanoshell (Zhu, Fang & Liu, 2020).
Through the open literature, from the best knowledge of authors, there is no investigation on the dynamics of aragonite doubly-curved nanoshell even at macro scale. Thus, the current work addresses a novel study on anisotropic nano-size structure. The nanoshell is modeled using a quasi-three-dimensional shell theory as a continuum model and the nonlocal strain gradient theory is used to catch up the small-size effects. The motion governing relations and associated boundary condition which is considered simply-supported in edges are solved analytically to find resonance position. Then, the numerical examples are expanded to show the sensitivity of resonance phenomenon to the nonlocality and strain gradient size-dependency, exponential factor, and geometrical shapes. A comparison between the present anisotropic model and its isotropic one is also performed.
Section snippets
Nonlocal strain gradient quasi-3D anisotropic shells formulation
This work is aim at investigating the forced resonance vibration of aragonite shell panels. Fig. 1 shows a doubly-curved shells in curvilinear coordinates (α, β, z) with thickness h, length a, and width b; radii of curvatures R1 (α, β) and R2 (α, β) along the two directions α and β. For a generic shell, we can write the kinematic, constitutive and equilibrium equations as follows (Chen, Wang, Hao & Zhang, 2017)
Solution procedure
In this subsection, an analytical based method using double-Fourier series is utilized to solve the problem. Hence, the displacements of the nanoshell should be selected as:herein denote the displacement amplitudes, Ω is exiting or load frequency, and ζ=mπ/a,
Numerical examples
This work aims at investigating the forced resonance vibration of doubly-curved nano-size shells made of aragonite with orthorhombic crystal system. Furthermore, using an exponential factor, the behavior of soft and stiff graded materials is analyzed. The numerical examples are also developed for different types of shell panels, (i.e. spherical (R1=R2=R), elliptical (R1=R, R2=2R), hyperbolic (R1=R, R2=-R), and cylindrical (R1=R, R2=→∞)). Fig. 2 shows different types of shell panels.
Concluding remarks
A non-isotropic mathematical size-dependent model was developed for studying the resonance phenomenon of a doubly-curved nanoshell made of aragonite material with orthorhombic crystal systems. To obtain the motion governing equations, the nonlocal strain gradient model in conjunction with a quasi-three-dimensional model was adopted. An analytical technique was performed to solve the problem for simply-supported edges. According to numerical examples, the conclusions of the current work can be
Declaration of competing interest
None.
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