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Research Papers

# Mechanics of Indentation into Micro- and Nanoscale Forests of Tubes, Rods, or Pillars

[+] Author and Article Information
Lifeng Wang

Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139

Christine Ortiz

Department of Materials Science and Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139

Mary C. Boyce

Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139mcboyce@mit.edu

J. Eng. Mater. Technol 133(1), 011014 (Dec 03, 2010) (9 pages) doi:10.1115/1.4002648 History: Received March 06, 2010; Revised August 09, 2010; Published December 03, 2010; Online December 03, 2010

## Abstract

The force-depth behavior of indentation into fibrillar-structured surfaces such as those consisting of forests of micro- or nanoscale tubes or rods is a depth-dependent behavior governed by compression, bending, and buckling of the nanotubes. Using a micromechanical model of the indentation process, the effective elastic properties of the constituent tubes or rods as well as the effective properties of the forest can be deduced from load-depth curves of indentation into forests. These studies provide fundamental understanding of the mechanics of indentation of nanotube forests, showing the potential to use indentation to deduce individual nanotube or nanorod properties as well as the effective indentation properties of such nanostructured surface coatings. In particular, the indentation behavior can be engineered by tailoring various forest features, where the force-depth behavior scales linearly with tube areal density ($m$, number per unit area), tube moment of inertia $(I)$, tube modulus $(E)$, and indenter radius $(R)$ and scales inversely with the square of tube length $(L2)$, which provides guidelines for designing forests whether to meet indentation stiffness or for energy storage applications in microdevice designs.

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## Figures

Figure 1

Schematic of nanoindentation tests on nanotube forests

Figure 2

(a) Schematic of indentation on nanotube forests and (b) diagram of a nanotube touched by the indenter

Figure 3

Schematic of penetration (contact) area on nanotube forests during indentation: A1—nanotubes are buckled; A2—nanotubes are under compression

Figure 4

Schematic of a column under compression force: (a) different boundary constraint in corresponding to the friction between indenter and the tubes and (b) typical force displacement relationship for a beam under axial compression

Figure 5

Indentation force for a single tube as a function of penetration depth based on nonlinear bending theory of a beam

Figure 6

(a) Comparison of different model predictions: Eq. 4 considering compression of tubes, Eq. 13 considering compression followed by elastic buckling of tubes, and Eq. 18 considering bending of tubes. (b) Magnified image at small indent depth. (c) Corresponding area A1 and A2 (i.e., number of tubes because m=1 tube/μm2 in this case) as a function of indent depth.

Figure 7

Comparison of energy absorption between different model predictions: Eq. 19 considering tube compression, Eq. 20 considering tube compression followed by elastic buckling, and Eq. 21 considering bending of tubes

Figure 8

(a) A two-dimensional square pattern of tube array and (b) a randomly distributed pattern of tube array

Figure 9

FEA results of load-depth behavior showing the effect of friction between the indenter and tubes

Figure 10

Comparison between analytical models and FEA results for indentation with indenter radius R=50–500 μm: (a) no friction between the indenter and nanotubes and (b) completely rough (no slipping) friction between the indenter and nanotubes. Top shows the corresponding displacement of tube forests at indent depth of h/L=0.1 with an indenter radius of 200 μm.

Figure 11

Evolution of number of tubes in contact with the indenter as a function of indent depth

Figure 12

Comparison between analytical models and FEA results for energy storage of indentation in Fig. 1: (a) no friction between the indenter and nanotubes and (b) completely rough friction between the indenter and nanotubes

Figure 13

Comparison of analytical models and FEA results with tube height L=10, 12.5, and 15 μm and indenter radius R=500 μm considering rough friction interface: (a) F versus h, (b) (F/Pcr) versus h, and (c) (F/Pcr) versus (h/L)

Figure 14

Comparison of analytical models and FEA results with indenter radius R=500 μm for different tube radius r=0.15−0.3 μm: (a) no friction between the indenter and nanotubes and (b) completely rough friction between the indenter and nanotubes

Figure 15

Comparison of analytical model and FEA results on the tilted angle of tubes: (a) no friction between the indenter and tubes and (b) completely rough friction between the indenter and tubes

Figure 16

Comparison of analytical models and FEA results with different areal density of tube forests (m=1/9,1/6,1/4 tubes/μm2) with indenter radius R=200 μm: (a) no friction between the indenter and tubes and (b) completely rough friction between the indenter and tubes

Figure 17

FEA results of indentation where tubes are randomly distributed as compared with the results for square distributed pattern: (a) no friction between the indenter and tubes and (b) completely rough friction between the indenter and tubes

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