A modified molecular-continuum model for estimating the strength and fracture toughness of graphene and carbon nanotube

A molecular-continuum model proposed previously for the estimation of elastic stiffness of nanomaterials is modified to estimate ultimate tensile strength and mode I/mode II fracture toughness of graphene and carbon nanotubes. By taking the modified Morse potential function and applying uniform strain field for a perfect specimen, a nonlinear stress-strain diagram can be plotted to estimate tensile strength. To estimate fracture toughness, a parameter called the strain intensity factor is introduced, and the near tip solution of linear elastic fracture mechanics rewritten in terms of strain intensity factor is used to locate the atoms of the cracked specimen. With the changes of bond distance and bond angle between atoms set in the deformed state, the potential energy within the region of representative volume is evaluated and treated as the strain energy in the cracked specimen. Using the well-known relation between strain energy release rate and stress intensity factor, a nonlinear generalized stress-strain diagram which showing the relation between stress intensity factor and strain intensity factor can be plotted. The estimated fracture toughness is then obtained from the maximum point of this diagram. To know whether our estimation is stable with respect to the crack size and tube radius, estimation based upon different parameters are presented for graphene and carbon nanotubes. By integrated symbolic and numerical computation, the results estimated by this model are shown to fall in the reasonable range predicted by the other experimental or numerical methods.