N. A. AlshammariN. ThamwattanaJ. A. McCoyD. BaowanB. J. CoxJ. M. HillSaudi Electronic UniversityUniversity of South AustraliaMahidol UniversityUniversity of Newcastle, AustraliaThe University of Adelaide2019-08-232019-08-232018-01-01Dynamics of Continuous, Discrete and Impulsive Systems Series B: Applications and Algorithms. Vol.25, No.5 (2018), 307-339149287602-s2.0-85052639067https://repository.li.mahidol.ac.th/handle/20.500.14594/46113© 2018 Watam Press. Numerous types of carbon nanostructure have been found experimentally, in-cluding nanotubes, fullerenes, nanocones and graphene. These structures have received much attention for their potential application in various nanoscale devices. The joining of different types of nanostructures may lead to further new structures with even more remarkable properties with possibly more potential applications. Using the calculus of variations, this paper models the join between different types of carbon nanostructures, namely carbon nanocone and fullerene, two nanocones, graphene and nanocone, fullerene and graphene, and nanocone and two parallel sheets of graphene. A perfect join configu-ration of absolute minimum energy is found in some circumstances. In general, the joining of these structures can be categorised into one of two models which are based on the cur-vature of the join profile. We refer to Model I when the join profile only involves positive curvature and Model II for the case of both positive and negative curvatures. We consider three cases of joining nanocones with a spherical cap: exactly half, more than half and less than half of a sphere. For two nanocones, we investigate the join between two symmetric carbon nanocones and the join between two nanocones which have different cone angles. These composite structures may be useful for the design of probes for scanning tunnelling microscopy and other nanoscale devices, such as carriers for drug delivery, and pores for gas and liquid separation. Copyright ?Mahidol UniversityMathematicsArticleSCOPUS