Analysis of fracture energy of composites reinforced by carbon nanotubes and carbon nanotube-grafted fibres using a non-linear bridging law

Carbon nanotubes can significantly increase the toughness of brittle and polymeric composites by bridging cracks and through pull-out from the matrix. The bridging law is essential for predicting the toughening mechanisms of the composites. Previous calculations of toughness are mostly based on tri-linear bridging laws. We present a non-linear smooth bridging law, and then apply it to calculate (i) the fracture energy of composites reinforced by carbon nanotubes, (ii) the pull-out force-displacement relation of a microscale carbon fibre that is grafted with carbon nanotubes on its surface, and (iii) the fracture energy of composites reinforced by grafted carbon fibres. The influence of various parameters on the fracture energies of the composites is revealed.