Elsevier

Engineering Structures

Volume 277, 15 February 2023, 115440
Engineering Structures

Vibrational characteristics of functionally graded graphene origami- enabled auxetic metamaterial beams with variable thickness in fluid

https://doi.org/10.1016/j.engstruct.2022.115440Get rights and content

Highlights

  • Genetic programming (GP)-assisted micromechanics models are used to predict the material properties.

  • Governing equations of the beam-fluid interaction system are discretised and solved by differential quadrature method.

  • The hydrodynamic effect of fluid on vibrational characteristics of novel FG-GOEAM beam is comprehensively investigated.

  • The FG-GOEAM beam outperforms its pristine metallic counterpart in terms of vibration performance.

Abstract

Within the framework of Timoshenko beam theory, this paper examines the effect of negative Poisson’s ratio (NPR) on the free vibration characteristics of functionally graded (FG) graphene origami (GOri)-enabled auxetic metamaterial (GOEAM) beams with variable thickness submerged in fluid. Poisson’s ratio and other elastic properties of the beam are graded in a layer-wise manner along the thickness direction and are predicted by genetic programming (GP)-assisted micromechanics models. With the hydrodynamic pressure effect on the beam being modelled as added mass, the equations of motion are derived from Hamilton’s principle then discretised and solved using the differential quadrature (DQ) method in conjunction with an iterative process to determine its natural frequencies and mode shapes. The effects of GOri folding degree, metamaterial distribution, graphene distribution pattern and content, fluid temperature and density on the free vibration behaviour of FG-GOEAM beams are discussed in detail. It is found that the FG-GOEAM beam outperforms its pristine metallic counterpart in terms of vibration performance.

Introduction

Composite structures that possess negative Poisson’s Ratio (NPR) present various desirable mechanical properties including superior vibration attenuation [1], enhanced impact resistance [2] and improved toughness [3]. With 3D isotropic materials exhibiting Poisson’s ratio values ranging between −0.5 and 1.0, this has garnered research momentum in the field of auxetic materials [4]. Though it has been determined that NPR behaviour can be engineered into the development of materials through the combination of atomic structures and various bonding types, nanomaterials like graphene are able to exhibit NPR due to the vacancy defects available within its structure [5]. This revelation lead to the advent of graphene origami (GOri) structures that possess ultra-high NPR.

The auxeticity of most of the existing metamaterials is highly dependent on their structural architecture/topology. These metamaterials are usually mechanically weak hence susceptible to localised deformation or failure under external loading, which leads to considerably deteriorated metamaterial properties. Graphene origami-enabled auxetic metallic metamaterials (GOEAMs) is a novel class of nanocomposites that simultaneously possess auxetic properties and improved mechanical behaviours. Schenk and Guest [6] presented two types of Miura-origami fold patterns: the folded structure, where opposite Poisson’s ratios are exhibited for in-plane and out-of-plane deformations within a single sheet; and the stacked folded Miura layers forming a 3D structure, resulting in a cellular folded auxetic material. Wei, et al. [7] investigated the in-plane and out-of-plane Poisson’s ratios in Miura-origami structures and demonstrated that the values are equal but opposite in sign, regardless of the material properties. Zhao, et al. [8] reported that GOri reinforcements embedded into a Cu matrix result in high NPR as well as improved elasticity behaviour of the nanocomposite. In addition, with a 3.35 wt% GOri composition arranged in the Miura pattern, an NPR of −0.2796 can be reached at room temperature. Kim, et al. [9] demonstrated the concept of a nanocomposite consisting of a single layer of graphene embedded in copper layers exhibiting ultra-high strengths of up to 4.0 GPa.

Functionally Graded (FG) nanocomposite structures reinforced with nanoparticles, such as graphene, are heterogeneous structures with material properties changing smoothly and continuously with respect to spatial coordinates. These structures exhibit exceptional mechanical performance because their material composition can be tailored to meet specific property requirements. Various studies reported that distributing more of the graphene nanofillers in the top and bottom layers of an FG structure increases its stiffness, subsequently endowing the structure with enhanced vibration resistance in beams [10], [11], [12], [13], [14], plates [15], [16], [17], [18] and shells [19]. Several studies have also demonstrated the effectiveness of a small addition of graphene to the matrix in terms of providing a drastic increase in the fundamental frequency of an FG structure [20], [21], [22]. Despite the numerous studies on the dynamic response of functionally graded structures, a substantial number are based on regular geometries, particular gradation types, specific boundary conditions and loading [23].

Understanding the interaction between fluid and structure is essential in the design of engineering systems across various fields, for instance, aircraft, floating structures and marine vessels. Li, et al. [24] analysed the hydroelastic free vibration behaviour of an FGM beam composed of aluminium and ceramic with variable thickness. Their results indicate that increasing the fluid density results in lower natural frequencies due to the higher added mass and hydrodynamic pressure. Wu, et al. [10] reported on the free vibration of an FG graphene reinforced composite beam partially immersed in a fluid and found that full submersion reduces the fundamental frequency of the beam. The free vibration of FGM plates in contact with fluid is also documented in various notable works [25], [26], [27], [28], [29], [30]. However, none of the above existing studies discussed the effect of NPR on the free vibration characteristics of functionally graded structures immersed in fluid.

The present study aims to fill this research gap to investigate the free vibration behaviour of functionally graded beams with variable thickness that are placed in fluid environment and made of graphene origami enabled metamaterials with tuneable NPR, with a particular focus on the effect of NPR on the natural frequencies and associated mode shapes of the beam. Theoretical formulations are derived within the framework of Timoshenko beam theory. The added mass theory is used to include the fluid effect on the FG-GOEAM beam structure. A modified Halpin-Tsai micromechanics model based on genetic programming (GP) algorithm developed by Zhao, et al. [31] is employed to predict the mechanical properties of the GOri metamaterial. Hamilton’s principle is utilised to derive the governing equations which are subsequently discretised through the use of the differential quadrature method (DQM). The natural frequencies and mode shapes of the FG-GOEAM beam in a fluid are ultimately computed via an iterative method. A parametric study on the influence of GOri folding degree, metamaterial distribution, GOri distribution pattern, GOri weight fraction, fluid density, and fluid temperature is conducted.

Section snippets

Functionally graded metamaterial beam

Consider an FG-GOEAM beam-fluid system as displayed in Fig. 1. The multilayer structure is characterised by a thickness h(z) and a length L, while immersed in fluid up to a level Lf. The origin of the 2D cartesian plane (y-z) is positioned on the midplane on one end of the beam with y and z denoting the horizontal and vertical axes, respectively. The beam thickness is governed by.hz=h01-μ0μz

whereby h0 is the averaged uniform thickness, μz depicts the variable thickness shape along the

Solution procedure

DQ method is used to solve the normalised equations of motion (Eq. 28) in conjunction with the corresponding boundary conditions (Eq. 29) and compatibility equations (Eq. 30). In this procedure, the derivative of a function at a particular point is approximated as the sum of linear weighted function values at all the sample points within a domain. This numerical procedure is highly efficient as accurate numerical solutions can be obtained through the use of a small number of grid points.

Validation and convergence

The validation of the current numerical procedure is conducted in comparison with the results provided by Li, et al. [24]. The variable thickness FGM beam in this study was composed of aluminium (Al) with Ea = 70 GPa, νa = 0.3, ρa = 2702 kg/m3, and ceramic (SiC) with Eb = 427 GPa, νb = 0.17, ρb = 3100 kg/m3. Additional properties of the beam include slenderness ratio η0 = 8, length of the beam L = 0.6 m, non-uniformity parameter μ0 = 0.2, power-law exponent n = 1 and fluid density ρf = 1000 kg/m

Conclusions

The free vibration characteristics of an FG-GOEAM beam immersed in fluid is explored within the framework of Timoshenko beam theory and Hamilton’s principle. A genetic programming based micromechanics model is applied to predict the elastic properties of the GOEAMs. The added mass is introduced to depict the fluid effect on the beam structure. The equations of motion of the beam-fluid interaction system are normalised by the DQM and solved via a direct iterative method. A parametric study with

CRediT authorship contribution statement

Bill Murari: Methodology, Investigation, Formal analysis, Validation, Writing – original draft. Shaoyu Zhao: Formal analysis, Writing – review & editing. Yihe Zhang: Resources, Writing – review & editing. Liaoliang Ke: Writing – review & editing. Jie Yang: Conceptualization, Funding acquisition, Supervision, Writing – review & editing.

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.

Acknowledgements

The work was fully supported by the Australian Research Council grant under the Discovery Project scheme (DP210103656). The authors are very grateful for its financial support.

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