3.22 | Spring 2008 | Graduate

Mechanical Behavior of Materials


Carbon Nanotube Mechanics

Group Members

  • Robert Mitchell
  • Mark Mascaro


Image removed due to copyright restrictions.
Please see: Kohlenstoffnanoroehre Animation



Pantano, Antonio, David M. Parks, and Mary C. Boyce. “Mechanics of Deformation of Single- and Multi-wall Carbon Nanotubes.” Journal of the Mechanics and Physics of Solids 52 (April 2004): 789-821.

Poncharal, Philippe, Z. L. Wang, Daniel Ugarte, and Walt A. de Heer. “Electrostatic Deflections and Electromechanical Resonances of Carbon Nanotubes.” Science 283 (March 5, 1999): 1513-1516.

Kyriakides, S., and Ju G. T. “Bifurcation and Localization Instabilities in Cylindrical Shells Under Bending–I. Experiments.” International Journal of Solids and Structures 29 (1992): 1117-1142.

Ju, G. T., and S. Kyriakides. “Bifurcation and Localization Instabilities in Cylindrical Shells Under Bending–II. Predictions.” International Journal of Solids and Structures 29 (1992): 1143-1171.

Wiki Assignments

Problem Set 2

(a) Pantano, et al. discuss several previously reported estimates of CNT elastic properties. Explain how the cited works in this review inferred the elastic properties of these CNTs from resonance of the structure (how did they do it, and how does resonance relate quantitatively to E?).

(b) On p. 795, the authors state that the CNT can be described with three elastic constants, but they also state that the wall material is isotropic. Isotropic materials require only 2 independent elastic constants, so demonstrate in terms of material and crystal symmetry why three are employed.

Problem Set 3

(b) Pantano, et al. extensively model the elastic buckling instabilities of CNTs, and essentially find that computationally expensive MD and FEM simulations predict the same critical loads as Timoshenko’s continuum elasticity theory for shells. In Pantano’s model of cantilevered CNT bending, they completely neglect the possibility of plastic deformation at the fixed end of the CNT cantilever. However, if the stresses get high enough in the CNT “beam”, plastic deformation will result. State the yield criteria / surface that you think best captures the onset of plastic deformation in a CNT (single or multiwalled; your choice), graph it as a yield surface under biaxial loading.

(c) Determine the deflection required to induce plastic deformation under cantilevered point loading. Be sure to clearly state the assumed fixed-end boundary conditions, and cite primary sources for any required data.

(d) Poncharal, et al. report the resonance of CNTs and demonstrate a frequency- and diameter-dependent response. Although this is structural resonance, it is similar in origin to the linear viscoelastic responses used to describe polymers. From this and other available sources, justify whether the CNT dynamics are best captured by a Maxwell, Kelvin-Voigt, or SLS model. Express the resonance of the CNTs in terms of the elements of the phenomenological spring/dashpot model you have chosen, keeping in mind that the elastic properties of the CNTs are well-known.

Problem Set 5

(b) The papers you have chosen discuss the elastic properties of CNTs. Of course, the applications people are excited about are related to the high tensile strength of these organic fibers. We know they’ll easily buckle in tension, and it’s hard to think of apps that would benefit from that. Read the 2 papers cited by Pantano, et al. that discuss nanotube fracture to review this concept.

Then compute the number of carbon bonds that must be broken to achieve tensile fracture of a single CNT (not a multiwall). It [elastic energy stored at fracture] should be the number of carbon bonds transversing the fracture plane, multiplied by the carbon-carbon bond strength, but is it?

Final Presentation

“Carbon Nanotube Mechanics.” (PDF)


Plasticity and fracture of microelectronic thin films/lines
Effects of multidimensional defects on III-V semiconductor mechanics
Defect nucleation in crystalline metals
Role of water in accelerated fracture of fiber optic glass
Carbon nanotube mechanics | Problem Set 2 | Problem Set 3 | Problem Set 5
Superelastic and superplastic alloys
Mechanical behavior of a virus
Effects of radiation on mechanical behavior of crystalline materials

Course Info

As Taught In
Spring 2008
Learning Resource Types
theaters Simulation Videos
grading Exams with Solutions
group_work Projects
assignment_turned_in Problem Sets with Solutions