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Research Papers

Theory on Bending in Cantilever Beams With Adsorbed Islands

[+] Author and Article Information
Chuangchuang Duan

LNM,
Institute of Mechanics,
Chinese Academy of Sciences,
Beijing 100190, China;
School of Engineering Sciences,
University of Chinese Academy of Sciences,
Beijing 100049, China
e-mail: duanchuangchuang@imech.ac.cn

Yujie Wei

LNM,
Institute of Mechanics,
Chinese Academy of Sciences,
Beijing 100190, China;
School of Engineering Sciences,
University of Chinese Academy of Sciences,
Beijing 100049, China
e-mail: yujie_wei@lnm.imech.ac.cn

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received April 14, 2017; final manuscript received May 20, 2017; published online May 31, 2017. Assoc. Editor: Kyung-Suk Kim.

J. Appl. Mech 84(7), 071006 (May 31, 2017) (7 pages) Paper No: JAM-17-1203; doi: 10.1115/1.4036819 History: Received April 14, 2017; Revised May 20, 2017

Traction between adsorbed islands and the substrate is commonly seen in both living and material systems: deposited material gathers into islands at the early stage of polycrystalline film deposition and generates stress due to lattice mismatch, cells exert cellular traction to extracellular matrix to probe their surrounding microenvironment in vivo, and so on. The traction between these islands and the substrate can result in perceivable macroscopic deformation in the substrate and may be measurable if the substrate is a cantilever beam. However, currently broadly used Stoney equation is incapable of handling such boundary condition. In this paper, we give the closed-form expression on the resulted curvature in substrate beams by distributed tractions. Such a relationship could be employed to monitor the stress evolution during thin film deposition, to quantify the stress level of cell traction as cells adhere to cantilever beams, and other related mechanical systems like charging–discharging induced stress in island-patterned electrode films. Moreover, we found that follower traction induced by an array of islands could lead to negative curvature. It shields light on the early stage compressive stress during polycrystalline film deposition.

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Figures

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Fig. 1

A cantilever beam with an array of islands bonded to its upper surface, where the lengths of the beam and one island are L and l, respectively. N denotes the number of islands and s denotes the length of the spacing between islands. τ refers to the maximum of the traction and EI refers to the flexural rigidity of the beam.

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Fig. 2

(a) A cantilever beam subjected to tangential force f(x), where the length of the beam is l and the eccentricity of the tangential force is e. (b) An element cut from the beam with sides normal to the deflected axis of the beam.

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Fig. 3

Schematic of the mth element consists of island and spacing: (a) free-body diagram of spacing between islands and (b) free-body diagram of the mth island

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Fig. 4

The curvature of the beam with patterned islands, where X denotes the coordinate from the fixed end of the beam (see Fig. 1): (a) curvature along the beam with 111 islands, (b) curvature within a period (with length 2πl6/τ¯ρ)), and (c) curvature within one island (with length l)

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Fig. 5

Nondimensional deflection curve of the beam: (a) the solid line stands for analytical solution and dots are FEM results, (b) ρ=0.5,N=4,20,100, ymax corresponds to the maximal deflection of the beam when ρ=0.5,N=4, (c) N=1000,ρ=0.05,0.1,0.15, ymax is the maximal deflection of the beam when ρ=0.15,N=1000, and (d) N decreases with increase in ρ, ymax refers to the maximal deflection of the beam when ρ=0.5,N=4

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