Modeling two-dimensional crystals and nanotubes with defects under stress

We calculate analytically the phase diagram of a two-dimensional planar crystal and its wrapped version with defects under external homogeneous stress as a function of temperature using a simple elastic square lattice model that allows for defect formation. The temperature dependence turns out to be very weak. The results are relevant for recent stress experiments on carbon nanotubes at high temperatures. Under increasing stress, we find a crossover regime which we identify with a cracking transition that is almost independent of temperature. Furthermore, we find an almost stress-independent melting point. In addition, we derive an enhanced ductility with relative strains before cracking between 200% and 400%, in agreement with carbon nanotube experiments. The specific values depend on the Poisson ratio and the angle between the external force and the crystal axes. We give arguments that the results for carbon nanotubes should be not much different from these results in spite of the different lattice structures.