Research Papers

Elastic Theory of Nanomaterials Based on Surface-Energy Density

[+] Author and Article Information
Shaohua Chen

Institute of Mechanics,
Chinese Academy of Sciences,
Beijing 100190, China
e-mail: chenshaohua72@hotmail.com

Yin Yao

Institute of Mechanics,
Chinese Academy of Sciences,
Beijing 100190, China
e-mail: yaoyin111@LNM.imech.ac.cn

1Corresponding author.

Contributed by the Applied Mechanics of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received July 9, 2014; final manuscript received October 6, 2014; accepted manuscript posted October 10, 2014; published online October 20, 2014. Editor: Yonggang Huang.

J. Appl. Mech 81(12), 121002 (Oct 20, 2014) (12 pages) Paper No: JAM-14-1302; doi: 10.1115/1.4028780 History: Received July 09, 2014; Revised October 06, 2014; Accepted October 10, 2014

Recent investigations into surface-energy density of nanomaterials lead to a ripe chance to propose, within the framework of continuum mechanics, a new theory for nanomaterials based on surface-energy density. In contrast to the previous theories, the linearly elastic constitutive relationship that is usually adopted to describe the surface layer of nanomaterials is not invoked and the surface elastic constants are no longer needed in the new theory. Instead, a surface-induced traction to characterize the surface effect in nanomaterials is derived, which depends only on the Eulerian surface-energy density. By considering sample-size effects, residual surface strain, and external loading, an explicit expression for the Lagrangian surface-energy density is achieved and the relationship between the Eulerian surface-energy density and the Lagrangian surface-energy density yields a conclusion that only two material constants—the bulk surface-energy density and the surface-relaxation parameter—are needed in the new elastic theory. The new theory is further used to characterize the elastic properties of several fcc metallic nanofilms under biaxial tension, and the theoretical results agree very well with existing numerical results. Due to the nonlinear surface effect, nanomaterials may exhibit a nonlinearly elastic property though the inside of nanomaterials or the corresponding bulk one is linearly elastic. Moreover, it is found that externally applied loading should be responsible for the softening of the elastic modulus of a nanofilm. In contrast to the surface elastic constants required by existing theories, the bulk surface-energy density and the surface-relaxation parameter are much easy to obtain, which makes the new theory more convenient for practical applications.

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Grahic Jump Location
Fig. 1

Reference, intermediate, and present configurations of a three-dimensional nanosolid with the corresponding volumes and surface areas V0 and S0, Vr and Sr, V and S, respectively. P is a surface traction that acts on the surface area SP, and f is a body force. The quantities u and ɛ are the displacement and strain fields, respectively.

Grahic Jump Location
Fig. 2

Schematic of a surface element in the reference, intermediate, and present configurations

Grahic Jump Location
Fig. 3

Schematic of an infinitesimal area element after deformation. The boundary lengths L1 and L2 change to become L1+ΔL1 and L2+ΔL2. ΔR is the normal displacement of the curved surface. The initial surface area A1 changes to become A1+dA1.

Grahic Jump Location
Fig. 4

Square-shaped nanofilm subjected to biaxial tension σ. The edge length is b and the thickness is h. As a result, the strain due to external biaxial tension is ɛ.

Grahic Jump Location
Fig. 5

Normalized biaxial moduli of different metallic nanofilms with {001} free surface as a function of film thickness: (a) copper, (b) silver, and (c) gold

Grahic Jump Location
Fig. 6

Normalized biaxial moduli of different metallic nanofilms with {111} free surface as a function of film thickness: (a) copper, (b) silver, and (c) gold

Grahic Jump Location
Fig. 7

Normalized biaxial moduli of different metallic nanofilms subjected to biaxial tension as a function of film thickness: (a) copper film with a {001} free surface and (b) silver film with a {111} free surface

Grahic Jump Location
Fig. 8

Schematic of a nanosolid surface. (a) Zero-thickness surface defined in surface elastic theory. (b) Top surface layer and a transition zone (physical surface) in atomistic simulations.



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