Research Papers

Failure Mechanics—Part II: The Central and Decisive Role of Graphene in Defining the Elastic and Failure Properties for all Isotropic Materials

[+] Author and Article Information
Richard M. Christensen

Professor Research Emeritus
Aeronautics and Astronautics Department,
Stanford University,
Stanford, CA 94305
e-mail: christensen@stanford.edu

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received August 14, 2014; final manuscript received August 18, 2014; accepted manuscript posted August 27, 2014; published online September 17, 2014. Editor: Yonggang Huang.

J. Appl. Mech 81(11), 111001 (Sep 17, 2014) (10 pages) Paper No: JAM-14-1370; doi: 10.1115/1.4028407 History: Received August 14, 2014; Revised August 18, 2014; Accepted August 27, 2014

Continuing from Part I (Christensen, 2014, “Failure Mechanics—Part I: The Coordination Between Elasticity Theory and Failure Theory for all Isotropic Materials,” ASME J. Appl. Mech., 81(8), p. 081001), the relationship between elastic energy and failure specification is further developed. Part I established the coordination of failure theory with elasticity theory, but subject to one overriding assumption: that the values of the involved Poisson's ratios always be non-negative. The present work derives the physical proof that, contrary to fairly common belief, Poisson's ratio must always be non-negative. It can never be negative for homogeneous and isotropic materials. This is accomplished by first probing the reduced two-dimensional (2D) elasticity problem appropriate to graphene, then generalizing to three-dimensional (3D) conditions. The nanomechanics analysis of graphene provides the key to the entire development. Other aspects of failure theory are also examined and concluded positively. Failure theory as unified with elasticity theory is thus completed, finalized, and fundamentally validated.

Copyright © 2014 by ASME
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Christensen, R. M., 2014, “Failure Mechanics—Part I: The Coordination Between Elasticity Theory and Failure Theory for all Isotropic Materials,” ASME J. Appl. Mech., 81(8), p. 081001. [CrossRef]
Christensen, R. M., 2013, The Theory of Materials Failure, Oxford University, Oxford, UK.
Cho, S., Chasiotis, I., Friedman, T. A., and Sullivan, J. P., 2005, “Young's Modulus, Poisson's Ratio and Failure Properties of Tetrahedral Amorphous Diamond-Like Carbon for MEMS Devices,” J. Micromech. Microeng., 15(4), pp. 728–735. [CrossRef]
Cho, S.-J., Lee, K.-R., Eun, K. Y., Hahn, J. H., and Ko, D.-H., “Determination of Elastic Modulus and Poisson's Ratio for Diamond-Like Carbon Films,” Thin Solid Films, 341(1–2), pp. 207–210. [CrossRef]
Robertson, J., 2002, “Diamond-Like Amorphous Carbon,” Mater. Sci. Eng., 37(4–6), pp. 129–281. [CrossRef]
Greaves, G. N., Greer, A. L., Lakes, R. S., and Rouxel, T., 2011, “Poisson's Ratio and Modern Materials,” Nat. Mater., 10(11), pp. 823–837. [CrossRef] [PubMed]
Spear, K. E., and Dismukes, J. P., 1994, Synthetic Diamond, Wiley, New York.
Chang, T., and Gao, H., 2003, “Size-Dependent Elastic Properties of a Single-Walled Carbon Nanotube Via a Molecular Mechanics Model,” J. Mech. Phys. Solids, 51(6), pp. 1059–1074. [CrossRef]
Popov, V. N., van Doren, V. E., and Balkanski, M., 2000, “Elastic Properties of Single-Walled Carbon Nanotubes,” Phys. Rev. B, 61(4), pp. 3078–3084. [CrossRef]
Tu, Z. C., and Yang, Z.-C., 2002, “Single-Walled and Multiwalled Carbon Nanotubes Viewed as Elastic Tubes With the Effective Young's Modulus Dependent on Layer Number,” Phys. Rev. B, 65(23), p. 233407. [CrossRef]
Li, C., and Chou, T.-W., 2003, “A Structural Mechanics Approach for the Analysis of Carbon Nanotubes,” Int. J. Solids Struct., 40(10), pp. 2487–2499. [CrossRef]
Treacy, M. M. J., Ebbesen, T. W., and Gibson, J. M., 1996, “Exceptionally High Young's Modulus Observed for Individual Carbon Nanotubes,” Nature, 381(6584), pp. 678–680. [CrossRef]
Zhao, P., and Shi, G., 2011, “Study of Poisson's Ratios of Graphene and Single-Walled Carbon Nanotubes Based on an Improved Molecular Structural Mechanics Model,” TechScience, 5(1), pp. 49–58. [CrossRef]
Scarpa, F., Adhikari, S., and Phani, A. S., 2009, “Effective Elastic Mechanical Properties of Single Layer Graphene Sheets,” Nanotechnology, 20(6), pp. 1–11. [CrossRef]
Al-Jishi, R., and Dresselhaus, G., 1982, “Lattice-Dynamical Model of Graphite,” Phys. Rev. B, 26(8), pp. 4514–4522. [CrossRef]
Lee, C., Wei, Z., Kysar, J. W., and Hone, J., 2008, “Measurement of the Elastic Properties and Intrinsic Strength of Monolayer Graphene,” Science, 321(5887), pp. 385–388. [CrossRef] [PubMed]
Mott, P. H., and Roland, C. M., 2009, “Limits to Poisson's Ratio in Isotropic Materials,” Phys. Rev. B, 80(13), p. 132104. [CrossRef]
Ely, R. E., 1965, “Strength of Graphite Tube Specimens Under Combined Stress,” J. Am. Ceram. Soc., 48(10), pp. 505–508. [CrossRef]


Grahic Jump Location
Fig. 1

Hexagonal symmetry for graphene

Grahic Jump Location
Fig. 2

Effective elastic member connecting carbon atoms

Grahic Jump Location
Fig. 3

2D ν2D* versus κ, Table 1

Grahic Jump Location
Fig. 4

3D ν* versus κ, Table 2

Grahic Jump Location
Fig. 5

Biaxial failure data for fused graphite, Ely [18]




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