A bridging law and its application to the analysis of toughness of carbon nanotube-reinforced composites and pull-out of fibres grafted with nanotubes

Bridging laws are essential in predicting the mechanical behaviour of conventional short-fibrereinforced composites and the emerging nanofibre-reinforced composites. In this paper, we first review some studies on the toughness of carbon nanotube-reinforced composites that is induced by the pull-out of the nanotubes from the matrix, and on the development of the corresponding bridging laws. A close examination of the available bridging laws for carbon nanotubes reveals that some fundamental issues need to be further addressed. We propose a simple nonlinear and smooth bridging law to describe the pull-out force–displacement behaviour of carbon nanotubes from a matrix. This law contains only two material parameters, reflects the basic features of the pull-out experiments, and is easy to use. We then use this bridging law to calculate the fracture toughness of carbon nanotube-reinforced nanocomposites and predict the pull-out force–displacement response of conventional short fibres that are grafted with carbon nanotubes. Some parametric studies are conducted to reveal the influence of various parameters at the nanoand micro-scale on these properties.