Vibrational characteristics of functionally graded graphene origami-enabled auxetic metamaterial beams based on machine learning assisted models

Abstract

This paper studies the free vibration behavior and dynamic responses of functionally graded (FG) beams made of novel graphene origami (GOri)-enabled auxetic metal metamaterials (GOEAMs) under an impulsive load within the framework of the first-order shear deformation beam theory. The auxetic property of the beam is effectively controlled by graphene content and GOri folding degree that are graded across the thickness direction of the beams in a layer-wise manner such that Poisson's ratio and other material properties are position-dependent and are estimated by the genetic programming (GP)-assisted micromechanical models. The governing equations are derived by using Lagrange equation approach together with Ritz method. Newmark-β method is employed to solve the governing equations for obtaining dynamic responses of the beams subjected to three different impulsive loads. A comprehensive parametric study is performed to examine the effects of graphene content, GOri folding degree and distribution patterns as well as temperature on the natural frequencies and dynamic responses of the beams. Numerical results show that high tunability in structural vibration characteristics can be achieved via GOri, which offers important insight into the application of FG-GOEAM beams in aerospace engineering structure for significantly improved dynamic structural performance.

Introduction

Metamaterials with auxetic characteristics have recently emerged as a novel type of advanced materials with unique mechanical properties that cannot be realized in conventional materials [1], [2]. For example, Auxetic metamaterials with mechanically tunable negative Poisson's ratio (NPR) were constructed through 3D hybrid double arrow-head structures [3], re-entrant hexagonal honeycombs [4], bi-material star-shaped re-entrant planar lattice structures [5], sinusoidally shaped beams [6], and planar structures connecting with anti-chiral topology [7]. These research efforts mentioned above were devoted to the design and property characterization of metamaterials only. Furthermore, the lattice/cellular metamaterials have been applied to sandwich composite structures to achieve unique and beneficial structural performances. Sandwich beam structures with auxetic cellular cores including truss and honeycomb were experimentally designed to investigate their bending and failure behaviors [8], [9]. The vibration and dynamic responses of sandwich composite structures with graded auxetic honeycomb and lattice cores were theoretical studied and found that the introduction of lattice/cellular metamaterials has remarkable influences on the structural behaviors [10], [11], [12]. However, the metamaterials used in the composite structures are cellular or lattice structures that are usually mechanically weak.

Inspired by Miura-origami metamaterial [13] and graphene origami (GOri) [14], a new class of high-performance GOri-enabled auxetic metallic metamaterials (GOEAMs) with tunable NPR and enhanced mechanical properties were developed [15], [16]. It is found that the tunability of the NPR can be achieved through changing graphene content, graphene folding degree, and temperature. Nevertheless, accurate evaluation of mechanical properties of the GOEAMs poses an enormous challenge. Halpin-Tsai model is a widely used theoretical model to estimate Young's modulus of graphene reinforced composites [17], [18], showing an excellent match with numerical and experimental results [15], [19]. Apart from the Halpin-Tsai model, another theoretical model named rule of mixture is frequently utilized to determine other key material properties including Poisson's ratio, coefficient of thermal expansion (CTE), and density of graphene reinforced composites [17]. However, these available theoretical models are simplified ones merely involving the influencing factors of graphene content and size, which cannot capture the effects of graphene folding degree and temperature. Therefore, we developed genetic programming (GP)-assisted micromechanical models to accurately estimate mechanical characteristics of the GOEAMs [20].

Functionally graded materials (FGMs) have received tremendous interest worldwide due to their excellent performances [21], [22]. Understanding the dynamic behaviors of composite structures made of FGMs is of importance for their practical applications in civil, mechanical, and aerospace engineering fields. Şimşek and Kocatürk [23] analyzed the effects of material gradient distribution on the natural frequencies and dynamic responses of a functionally graded (FG) beam under a point moving load. Chen et al. [24] designed novel FG porous beams with symmetric and asymmetric porosity distribution patterns, and studied their free and forced vibration features. Moreover, Tu et al. [25] investigated the vibration characteristics of FG plates within the framework of higher order shear deformation theory and Navier's solution procedure. The free vibration characteristics of FG porous micro-plates are further studied by Kim et al. [26] employing the Kirchhoff and Mindlin plate theories whose microstructure effects are captured by means of the modified couple stress theory. The mechanical behaviors and vibration characteristics of various FGM structures such as plates and curved shells were widely investigated as well based on higher order shear deformation theories [27], [28], [29], [30], [31], [32], [33], [34], [35], [36].

With the introduction of graphene nanofillers, Kitipornchai et al. [37], Feng et al. [38], and Song et al. [39] proposed FG graphene platelets reinforced metal porous composite beams, polymer composite beams, and polymer composite plates, respectively to investigate their vibration performances and manifested that the incorporation of graphene can significantly strengthen the FG composite structures with the natural frequencies being increased. Thai et al. [40] utilized the non-uniform rational B-spline formulation to study the free vibration properties of FG graphene platelets reinforced composite plates whose Young's modulus was estimated by the Halpin-Tsai model. On the basis of a Mori-Tanaka micromechanical model and finite element method, García-Macías et al. [41] analyzed free vibrations of FG composite plates reinforced by graphene. Wang et al. [42] further investigated the vibration responses of FG graphene reinforced beams under two successive moving masses based on a new high order shear deformation theory. In addition, the strengthening effects of graphene platelets on vibration characteristics were comprehensively investigated for various structures such as nanoplates [43], [44], composite plates [45], eccentric annular plates [46], porous plates [47], [48], dielectric plates [49], arches [50], cylindrical panel [51], [52], cylindrical shells [53], [54], shallow shells [55], disk-shaft assembly [56], and pre-twisted blade-shaft system [57] as well. Despite the superior reinforcing effect, the research work mentioned above failed to take into consideration the effect of NPR caused by the GOEAMs.

Combining FGMs with our developed GOEAMs contributes to the development of novel FG-GOEAM structures featured with the gradient distributions of GOri content and GOri folding degree, enabling the Young's modulus and Poisson's ratio to be design variables to realize enhanced structural performances [20]. This study aims to analyze the free vibration and dynamic characteristics of such FG metamaterial composite beams subjected to impulsive loads. The GP-assisted micromechanical models are adopted to determine the material properties such as Young's modulus, Poisson's ratio, CTE, and density of the metamaterials. The governing equations are obtained from the theoretical derivation by means of the Timoshenko beam theory, Lagrange equation, as well as Ritz method. The effects of graphene content, graphene folding degree, and graphene distribution patterns are investigated in detail to find the most effective way to enhance the structural dynamic properties of the FG-GOEAM beams.