Tunable nonlinear bending behaviors of functionally graded graphene origami enabled auxetic metamaterial beams

Abstract

This paper investigates tunable nonlinear bending behaviors of functionally graded composite beams made of graphene origami (GOri)-enabled auxetic metal metamaterials (GOEAMs) within the theoretical framework of the first-order shear deformation theory and von Kármán type nonlinearity. The beam is comprised of multiple GOEAM layers with GOri content and folding degree being variables to effectively control its auxetic property that is graded from layer to layer across its thickness direction. Our developed genetic programming (GP)-assisted micromechanical models are used to estimate the position- and temperature-dependent Poisson’s ratio and other material properties of each GOEAM layer in the beam. The nonlinear governing equations of the FG-GOEAM beam are derived by the principle of virtual work and numerically solved by the differential quadrature (DQ) method. A detailed parametric investigation is conducted to examine the effects of GOri content, folding degree, and temperature on the tunability of the nonlinear bending deflection and normal stress of the FG metamaterial beam. Numerical results offer significant insights into the design of FG-GOEAM beam structures with enhanced bending performances.

Introduction

Metamaterials are a novel type of advanced materials that possess unusual and sometimes even unprecedented physical and mechanical properties. During the last few decades, developed electromagnetic/photonic/acoustic metamaterials allow the realization of the negative index in light refraction, enhanced optical nonlinearity, and cloaking devices [1], [2], [3]. More recently, the concept of metamaterial has also been carried over to mechanical fields. Negative Poisson’s ratio (NPR) is a typical mechanical metamaterial characteristic that is hard to be achieved in conventional materials. Lakes [4] first reported a novel foam structure with NPR. Bückmann et al. [5] fabricated a three-dimensional mechanical metamaterial assembled by basic bow-tie elements to achieve NPR using the dip-in direct-laser-writing optical lithography technique. Chen et al. [6] designed a class of architected lattice metamaterials that exhibit tunable NPR and vibration-mitigation capability. More auxetic metamaterials were constructed by connecting planar structure with anti-chair topology [7], introducing a regular array of holes in common metals and plastics [8], and assembling wedge-shaped re-entrant unit cells [9]. However, these aforementioned auxetic metamaterials were designed based on lattice or cellular structures that are usually mechanically weak. Besides, these research works were only devoted to the material design and its property characterization.

Auxetic metamaterials have been applied to cores in sandwich composite structures to realize unique structural properties. Hou et al. [10] experimentally and numerically investigated the bending and failure behaviors of sandwich beams with auxetic cellular cores and found that the gradient core has a remarkable effect on the bending performances of the sandwich structures. Li and Wang [11] further performed three-point bending tests to study the bending properties of sandwich structures with 3D printed truss, conventional honeycomb, and re-entrant honeycomb cores. It is concluded that the sandwich composite structures with re-entrant honeycomb core possess superior energy absorption capability but lower strength and stiffness than the other two cases. Moreover, vibration and dynamic characteristics of sandwich composite structures with functionally graded auxetic 3D lattice core [12] and 3D chiral auxetic core [13] were studied, respectively. It is found that the sandwich metamaterial composite beam with pyramidal lattice core exhibits vibration insulation performance [14]. Again, the metamaterials used in the composite structures are still lattice/cellular structures that would weaken the structural stiffness.

With the inspiration of Miura-origami metamaterials [15] and graphene origami (GOri) structures [16], Zhao et al. [17], [18] recently designed a new class of GOri-enabled auxetic metal metamaterials (GOEAMs) with highly programmable NPR and negative thermal expansion as well as enhanced mechanical properties employing molecular dynamics (MD) method. The NPR and other mechanical properties of the metamaterials can be effectively tuned by the GOri content, folding degree, and temperature. Furthermore, more properties including interfacial shear strength [19] and toughness [20] of the materials were systematically studied by them as well. In order to accurately and efficiently predict the material properties such as NPR, Young’s modulus, mass density, and coefficient of thermal expansion (CTE) of the proposed metamaterials, the authors further developed genetic programming (GP)-assisted micromechanical models [21].

Composite structures made of functionally graded materials (FGMs) that are characterized by the continuous variation of material components or properties across one direction have been paid broad attention over the world because of their advantageous properties. Reddy [22] analyzed the functionally graded (FG) plates based on third-order shear deformation theory. Neves et al. [23] investigated static bending, free vibration and elastic buckling of FG plates with the theoretical framework of higher-order shear deformation theory and a meshless technique. Chen et al. [24], [25] developed a kind of FG porous beams and studied their static bending, elastic buckling, as well as free and forced vibration performances. By introducing graphene into FGMs, Yang et al. [26] first proposed FG graphene reinforced composite structures that exhibit significantly enhanced structural properties. The nonlinear bending [27], [28], nonlinear free vibration [29], [30], buckling and postbuckling [31], [32] of the FG graphene reinforced composite beams were comprehensively studied and concluded that the addition of a small account of graphene can greatly improve the beam stiffness with their bending deflection, natural frequency, and buckling resistance being enhanced. Moreover, numerous researchers carried out a lot of research on the structural behaviors of FG graphene reinforced composite structures as well [33], [34], [35], [36], [37], [38], [39]. The aforementioned research work, however, failed to take into account the effect of NPR of GOEAMs on the structural properties.

With the combination of FGMs and the designed GOEAMs, we present a multilayer FG-GOEAM beam featured with a layer-wise graded variation of GOri content along its thickness direction. The objective of this paper is to investigate the nonlinear bending behaviors of the FG-GOEAM beam tuned by GOri material parameters and distributions. The material properties including Poisson’s ratio, Young’s modulus, mass density, and CTE of GOEAMs in each individual layer of the FG metamaterial beams are estimated by the developed GP-assisted micromechanical models. The nonlinear governing equations are derived within the theoretical framework of first-order shear deformation theory and von Kármán type geometric nonlinearity. Differential quadrature (DQ) method is adopted to numerically solve the governing equations to obtain the bending deflection and normal stress. A parametric study is carried out with a focus on the effects of GOri content, folding degree, and temperature on tuning the nonlinear bending behaviors of the FG beams.