  We calculate analytically the phase diagram of a two-dimensional square
crystal and its wrapped version with defects under external homogeneous stress
as a function of temperature using a simple elastic 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. 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 200-400%, 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 are not much different to the
wrapped square crystal.
