A recently developed approximate approach for modeling of carbon nanotubes and for evaluation of the effective properties of carbon nanotubes reinforced materials is applied to analysis of square plate with hole under uniaxial tension. It is assumed, that the plate is made of polymer with randomly distributed and randomly oriented carbon nanotubes. Distributions of carbon nanotubes volume fraction is varied over the surface of the plate, with the constraint that the total volume of nanotubes in the structure is the same, and the effects of that on stress concentration is evaluated. Such special variation volume fraction makes the carbon nanotubes reinforced polymer a functionally graded material. In evaluation of the local effective properties the Gutrin–Murdoch material surface model is applied to a hollow cylinder of finite thickness and combined with the notion of energy equivalence to replace carbon nanotube with an equivalent solid cylinder. The interphase, if any, can be modeled as a spring layer and subsequently added to obtain another equivalent solid cylinder. The material reinforced with such solid cylinders is then evaluated using the method of conditional moments, a stochastic homogenization approach proposed earlier for determination of the effective properties of random materials.
Bibliographical noteFunding Information:
LN and HS gratefully acknowledge the financial support by the German Research Foundation (DFG) via Projects NA1203/1-1 and NA1203/1-2 .
- Carbon nanotubes
- Functionally graded material
- Gurtin–Murdoch surface model
- Spring layer model
- Statistical method