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Research Papers

Thin-Shell Thickness of Two-Dimensional Materials

[+] Author and Article Information
Enlai Gao

Applied Mechanics Laboratory,
Department of Engineering Mechanics,
Center for Nano and Micro Mechanics,
Tsinghua University,
Beijing 100084, China

Zhiping Xu

Applied Mechanics Laboratory,
Department of Engineering Mechanics,
Center for Nano and Micro Mechanics,
Tsinghua University,
Beijing 100084, China;
State Key Laboratory of Mechanics and
Control of Mechanical Structures,
Nanjing University of
Aeronautics and Astronautics,
Nanjing 210016, China
e-mail: xuzp@tsinghua.edu.cn

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received August 21, 2015; final manuscript received September 5, 2015; published online October 1, 2015. Editor: Yonggang Huang.

J. Appl. Mech 82(12), 121012 (Oct 01, 2015) (4 pages) Paper No: JAM-15-1444; doi: 10.1115/1.4031568 History: Received August 21, 2015; Revised September 05, 2015

In applying the elastic shell models to monolayer or few-layer two-dimensional (2D) materials, an effective thickness has to be defined to capture their tensile and out-of-plane mechanical behaviors. This thin-shell thickness differs from the interlayer distance of their layer-by-layer assembly in the bulk and is directly related to the Föppl–von Karman number that characterizes the mechanism of nonlinear structural deformation. In this work, we assess such a definition for a wide spectrum of 2D crystals of current interest. Based on first-principles calculations, we report that the discrepancy between the thin-shell thickness and interlayer distance is weakened for 2D materials with lower tensile stiffness, higher bending stiffness, or more number of atomic layers. For multilayer assembly of 2D materials, the tensile and bending stiffness have different scaling relations with the number of layers, and the thin-shell thickness per layer approaches the interlayer distance as the number of layers increases. These findings lay the ground for constructing continuum models of 2D materials with both tensile and bending deformation.

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Figures

Grahic Jump Location
Fig. 1

(a) Tensile stiffness K, bending stiffness D, (b) D/K ratio, and the Föppl–von Karman number γ calculated for 2D materials. For γ, we consider a typical size of 2D materials of 10 μm. The dashed lines in panel (a) indicate the boundaries of D/K ratios, and the solid/dashed lines in panel (b) are theoretical predictions with ν = 0.3 based on Eqs. (1) and (2).

Grahic Jump Location
Fig. 3

(a) The relation between thin-shell thickness ts, interlayer distance d, and tensile stiffness K of 2D materials, where the value of ts approaches d with decreasing K. (b) The relation between ts, d, and the bending stiffness D for graphene multilayers. The value of ts approaches d as the number of layers N increases, due to the faster increase of D than that of K as a function of N (data taken from Ref. [36]).

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