An accurate size-dependent sinusoidal shear deformable framework for GNP-reinforced cylindrical panels: Applications to dynamic stability analysis
Introduction
Polymer is an adequate candidate of fabricating microsensors and microactuators with high efficiency and low energy consumption. Adding carbon-based nanofillers can improve the mechanical, thermal and electrical properties of polymers on the premise of retaining its easy machining performance and make the polymer-based devices more applicable in some special cases charactering low frequency, low force generation, and short dynamic range [[1], [2], [3], [4]]. As new type of materials incorporating multi-phase solids, more and more scientists devoted efforts to apply them to design the micro/nano-scale devices, such as micro/nano-robots for drug delivery, nano-sensors, nano-actuators, nano-transducers and biosensors [5,6]. The carbon nanotube (CNT) [[7], [8], [9]], graphene nanoplatelet (GNP) [10,11] and graphene oxide (GO) [12,13] can be used as reinforcement nanofillers to improve the mechanical properties of polymers. Graphene nanoplatelet is proved to be one of the most ideal nano-reinforcements for polymer composite owing to its excellent inherent natures [[14], [15], [16], [17]]. Very recently, the marriage of GNP reinforced polymer composite and functionally graded material (FGM) provides an effective strategy of sufficiently using the GNPs and polymers and creates the functionally graded graphene nanoplatelet reinforced composite (FG-GNPRC), in which the weight fraction of GNP nanofillers varies gradually from one surface to a preferred one [[18], [19], [20], [21], [22]]. It is worth emphasizing that the fabrication of such functionally graded structures with a continuous and smooth variation of GNPs is extremely difficult due to the constraint of current manufacture technology. For overcoming this problem, the functionally graded GNP-reinforced multilayer structures are introduced as an excellent alternative [23,24]. A functionally graded GNP reinforced multilayer nanocomposite structure that stacked up with a number of individual layers, in which GPN weight fraction remains constant within each layer but follows a layer-wise gradual change through thickness, is much easier to fabricate and has been proved to be an excellent approximation to the ideal functionally graded structure with a continuous and smooth variation of GNPs when the total number of layers is sufficiently large [[25], [26], [27]]. As a novel advanced composite material having excellent mechanical properties, FG-GNPRCs have received increasing attentions from researchers due to their potential applications in different engineering fields [[28], [29], [30], [31]].
Beam, plate, shell and panel in small scales are all fundamental structural components that widely used in the micro- and nano-electromechanical devices (MEMs and NEMs), predicating their mechanical performance is of importance for the engineers and researchers. It is well known that due to the small-scale effects size-dependent behaviors of microstructures take place and thus the classical elasticity theories is not able to handle the mechanical response of these small-scale structures. The modified couple stress theory (MCST) is a simply and efficient size-denpendent theory that contains only one material length scale parameter. The physical meaning of the material length scale parameters is probably that it describes a representative size of strong strain gradient effect zone near interface, or surface, or singular points when material is loaded, it should be the material parameter [32,33]. Moreover, MCST has been extensively implemented in the analysis of FG microbeams, plates and shells [[34], [35], [36], [37], [38]]. The mechanical responses of FG-GNPRC components at micro-/nano-scales have also been reported in published literatures. Arefi et al. [39,40] performed investigations of bending and vibration response of the FG-GNPRC curved nanobeams and nanoplates based on a nonlocal elasticity theory. Sahmani and Aghdam [41] developed a nonlocal strain gradient beam model and investigated the nonlinear vibrations a pre-buckled and post-buckled FG-GNPRC nanobeam. Tsiatas and Yiotis [42] examined size effect on orthotropic Kirchhoff-type skew micro-plates based on a modified couple stress theory. Thai et al. [43] proposed a size dependent computational model based on the modified strain gradient theory and higher-order shear deformation theory and researched free vibrations of FG-GNPRC microplates. In addition, further literature review indicates that most of the existing contributions focus on FG-GNPRC micro-beams [44,45], micro-plates [46,47] and micro-shells [48,49]; nevertheless very limited works reported the mechanical behaviors of FG-GNPRC cylindrical micro-panels. The cylindrical panels always suffer external dynamic forces owing to their light weight and excellent properties in terms of dynamic behavior, strength and stability. Hence, proper understanding the dynamic behaviors of the micropanels is essential toward development of reliable models for different types of MEMs and NEMs.
The current work, for the first time, focus on the dynamic stability of functionally graded graphene nanoplatelets reinforced composite (FG-GNPRC) cylindrical micro-panels subjected to an axial oscillation compression. As another main contribution, the present work developed an accurate, modified couple stress-based cylindrical micro-panel model within the framework of sinusoidal shear deformation theory (SSDT), in which the classical strain tensors and curvature tensor components are given with consideration of the geometric curve of cylindrical panels. The Mathieu-Hill equations of governing the dynamic stability behaviors are yielded based on the proposed panel model and Lagrange's equations, and the equations are solved using Bolotin's method. Prior to stability analysis, the free vibration and buckling responses of functionally graded (FG) cylindrical panels with simply-supported boundary condition are calculated and compared with the published results, verifying the validation of proposed methodology. The modified Halpin-Tsai micromechanical modeling is implemented to determine the effective Young's modulus of FG-GNPRC, while the mass densities and Poisson's ratios are calculated using the rule of mixture. Some parameter studies are performed and the results in graphical form are presented to show the effects of dimensionless length parameter, GNP distribution patterns, GNP size and shape, length-to-span ratio and radius-to-span ratio of cylindrical panels on size-dependent dynamic stabilities of the FG-GNPRC cylindrical micro-panels.
Section snippets
Description of problem
An FG-GNPRC cylindrical micro-panel under axial oscillation compression is taken into account. Fig. 1 schematically depicts the problem under consideration. It is assumed that the uniform distributed load acts upon the mid-plane; N(t) is the axial oscillation compression load. (x, θ, z) is a cylindrical coordinate system fixed at the mid-surface of the micro-panel, and u, v, and w are the displacement components in the x−, θ−, and z− directions. The symbols R, h, and L represent radius,
Modified couple stress-based sinusoidal shear deformation cylindrical micro-panel model
There are various shell models developed based on different shear deformation theories [[52], [53], [54], [55]]; however, the literatures reported modified couple stress-based high order shear deformable models are still limited. As a first endeavor, the current work develops a sinusoidal shear deformation cylindrical panel model based on the modified couple stress theory. On the basis of sinusoidal shear deformation theory, the displacement fields (u, v, w) of a cylindrical panel are given as
Comparison research
There are limited literatures of reporting the mechanical behaviors of FG cylindrical micro-panels, thus being lack of numerical results for direct comparison purpose. Therefore, the free vibrations and buckling responses of FG plates in micro scales or cylindrical panels in macro scales are considered as special cases of cylindrical micro-panel to check the validation of the stiffness and mass matrixes derived in present work. In this section, the power-law function and
Parameter studies
This section gives some numerical examples to show the dimensionless length scale parameter, GNP distribution patterns, GNP size and shape, length-to-span length ratio of panel and radius-to-span length ratio of panel on the size-dependent stability behaviors of the panel.
In the following, unless otherwise stated, 1 the dimensions of graphene nanoplatelets are lGNP = 2.5 nm, wGNP = 1.5 nm, and hGNP = 0.3 nm. The material properties of GNPs and epoxy are ρGNP = 1.06 g/cm3, EGNP = 1.01 TPa, ρM
Conclusions
A modified couple stress-based sinusoidal shear deformable cylindrical micro-panel model is developed to deal with size-dependent dynamic stability behaviors of the FG-GNPRC cylindrical micro-panels subjected to an axial oscillation compression. The classical strain tensors and curvature tensor components are presented with consideration of geometric curve of cylindrical panels. A general Lagrange procedure is implemented and yields a system of Mathieu-Hill equations of governing dynamic
Author statement
Yuewu Wang: Conceptualization, Methodology, Writing-Original draft preparation. Tairan Fu: Software, Validation, Writing-Reviewing, Language. Wei Zhang: Supervison, Conceptualization, Methodology.
Declaration of competing interest
The authors declare no conflicts of interest.
Acknowledgements
The authors gratefully acknowledge the support of National Natural Science Foundation of China (NNSFC) through Grant Nos. 11832002, 11427801 and 51976097, and the Funding Project for Academic Human Resources Development in Institutions of Higher Learning under the Jurisdiction of Beijing Municipality (PHRIHLB).
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