Elsevier

Acta Materialia

Volume 228, 15 April 2022, 117791
Acta Materialia

Full length article
Genetic programming-assisted micromechanical models of graphene origami-enabled metal metamaterials

https://doi.org/10.1016/j.actamat.2022.117791Get rights and content

Abstract

Graphene origami (GOri) enabled metallic metamaterials are novel nanomaterials simultaneously possessing negative Poisson's ratio (NPR) and enhanced mechanical properties that are independent of the topology/architecture of the structure. Predicting their material properties via existing micromechanical models, however, is a great challenge. In this paper, a highly efficient micromechanical modeling approach based on molecular dynamics (MD) simulation and genetic programming (GP) algorithm is developed to address this key issue. The GP-based Halpin-Tsai model is extensively trained from MD simulation data to accurately predict the Young's modulus of GOri/Cu metamaterials with various GOri folding degrees, graphene contents and temperatures with a high coefficient of determination (R2) of ∼0.95. Meanwhile, the well-trained GP-based rule of mixture can accurately predict the coefficient of thermal expansion (CTE), Poisson's ratio and density of metamaterials with R2 of ∼0.95, ∼0.93 and ∼0.99, respectively. The excellent agreement between our estimated results and experimental data shows that the models developed herein are highly efficient and accurate in predicting mechanical properties that are essential for the analysis and design of functionally graded metal metamaterial composite structures. The theoretical results demonstrate that the proposed functionally graded metamaterial beam achieves significantly improved bending performance.

Introduction

Mechanical metamaterials with negative Poisson's ratio (NPR) have attracted considerable attention worldwide due to their unique and advantageous physical and mechanical properties such as outstanding fracture toughness, indentation hardness, energy absorption capability, shear resistance, and many others [1], [2], [3], [4]. These make them very promising in many engineering applications in aerospace, mechanical, automotive, and defense areas [1]. It should be noted that there are no auxetic metamaterials in nature. Most of them are in fact not materials but man-made cellular or lattice structures made of natural materials with appropriately designed architecture/topology to exhibit auxetic features [5], [6], [7]. These structures are usually mechanically weak and susceptible to excessive deformation or local failure that may lead to the deterioration or even loss of the auxetic behavior. Therefore, it is of great significance to develop metamaterials with NPR characteristics and enhanced mechanical properties that are independent of the architecture/topology.

Since the discovery of graphene in 2004, it has received wide research interest since it possesses outstanding physical properties. It also shows tunable NPR features [8] that can be realized via defect engineering [9], functionalization [10], and origami design [11]. Due to its excellent mechanical properties, graphene has been widely used as the reinforced filler for polymer, metal, or ceramic matrices to achieve high-performance composite materials [12,13]. Lots of experimental investigations on mechanical properties of graphene reinforced metal matrix composites were carried out [14,15], but most research works focused on only one or two variables of the samples, e.g., graphene content or size. Furthermore, the inclusion of multiple influencing factors in a single experimental study requires greater efforts and is more time-consuming than that in numerical simulations. To overcome the limitation of experiments, we have employed molecular dynamics (MD) simulations to investigate the interfacial performance [16], strengthening and toughening property, as well as metamaterial characteristic [17] of graphene reinforced copper nanocomposites considering different influencing factors. With the aid of graphene origami (GOri), we designed a kind of metallic metamaterials that exhibit NPR and enhanced mechanical properties simultaneously [17]. However, MD simulation becomes computationally expensive for large-scale GOri-metallic metamaterials. In this case, it is ideal to use theoretical models, such as the widely used Halpin-Tsai model, to predict the mechanical properties of the metamaterials. Halpin-Tsai model is able to predict Young's modulus of graphene reinforced composites [18,19] in good agreement with experimental results [20]. However, the theoretical model is generally simplified by merely considering the influence of graphene content and its size, which leads to inaccurate results when graphene content is greater than a threshold value [20]. Other important factors such as GOri folding degree or temperature are not included in this model that cannot predict Young's modulus of the metamaterials, limiting its more extensive application in practical engineering. Therefore, it still remains a great challenge to evaluate the material properties of GOri-enabled metallic metamaterials with various influence factors.

The rise of machine learning (ML) opens up a new route to predict material properties [21,22]. One can readily obtain the target values of composites with any combination of influence factors within a minute using well-trained ML models. Artificial neural network (ANN) is the most widely used ML algorithm for material performance prediction [23], [24], [25]. Support vector machine (SVM) is also a powerful tool to predict material properties [26]. These approaches, however, are difficult to develop an explicit mathematical relation between the input variables and output results. Genetic programming (GP) is a superior ML algorithm that is capable of generating simplified prediction expressions [27], overcoming the shortcomings of the aforementioned approaches. Gandomi and Alavi [28] adopted the GP method to model material properties and structural behaviors of the nonlinear systems and demonstrated that the approach is reliable and accurate.

Aiming to address the above challenges, we develop GP-based micromechanical models for accelerated prediction of mechanical properties of the GOri-enabled metallic metamaterials. At first, the mechanical parameters including Young's modulus, coefficient of thermal expansion (CTE), Poisson's ratio, and density of the metamaterials with different graphene folding degrees (H coverages on the creases), graphene contents, and temperatures are obtained using MD method [17,29]. Afterwards, these MD data are trained via GP algorithm to acquire the well-trained function expressions to modify the Halpin-Tsai model and rule of mixture. Finally, the developed GP-based micromechanical models are used for the static bending analysis of functionally graded metal metamaterial beams.

Section snippets

MD simulation

The MD simulations were performed using the LAMMPS package [30]. The embedded-atom method (EAM) potential [31], AIREBO potential [32], and LJ potential were adopted to describe the noncovalent interactions of Cu atoms, covalent interactions of C-C atoms and C-H atoms for the GOri, as well as vdW interactions between graphene and Cu matrix, respectively. The detailed potential parameters used in our simulation can be found in Refs. [16,17,33]. The GOri was designed with the aid of hydrogenation.

Mechanical properties of graphene origami-enabled metal metamaterials

Dataset of mechanical properties of the metamaterials is generated based on extensive MD simulations. Here, we choose single-crystal copper (Cu) with a face-centered-cubic (FCC) lattice structure as a matrix of metallic metamaterials. The polycrystal nature of copper is also investigated in Fig. S1 (Supplementary Information) in which the Young's modulus of the single-crystal model is very close to that of the polycrystal model. The atomic configuration of GOri is built by chemisorbing hydrogen

Genetic programming based on MD simulations

We establish the ML model based on the GP algorithm and MD simulation data to find the mapping relationship between the input features and output targets. From the MD simulation results in the previous section, H coverage, graphene content, and temperature are key factors to affect the mechanical properties of graphene reinforced composites. The size of graphene is not that important for the mechanical behaviors of composite structures [19,39]. As a result, we only consider the three factors (H

Static bending behaviors of functionally graded metal metamaterial beams

Functionally graded metamaterials are inhomogeneous metamaterials with gradient properties by progressively varying the material composition and/or microstructure along one or more directions. A level set-based topology optimization method was developed to design the functionally graded metamaterials with auxetic properties [48]. Kuang et al. [49] proposed a grayscale digital light processing 3D printing method to fabricate functionally graded metamaterials with negative Poisson's ratio.

Conclusions

To conclude, we developed new micromechanical models for the predictions of material properties of GOri-enabled metal metamaterials using MD simulation and GP approach. The features of 6 H coverages, 8 graphene contents and 8 temperatures were selected to build the GOri/Cu metamaterials and perform the MD simulations. Our simulation results suggest that the maximum NPR of GOri/Cu is –0.59 when the HGr is 100%, VGr is 14.47 vol%, and T reaches 800 K with an improved elastic modulus in comparison

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.

Acknowledgments

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

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