  The abrupt loss of mechanical stability of two-dimensional graphene-type
crystals at a certain transition temperature is described. At this temperature,
the graphene state with practically zero-speed bending sound and developed
bending fluctuations becomes energetically favorable. Such phenomenon, akin to
melting, is naturally caused by the anharmonicity of crystal oscillations. In
order to circumvent the known difficulties associated with taking into account
the anharmonic effects, we propose an original pseudo-harmonic approximation,
within which we determine the free energy of the anharmonic crystal and find a
numerical characteristic for the intensity of bending vibrations at transition
temperature. This characteristic is similar to the empiric Lindemann criterion
for the melting phenomenon. At the same time, in contrast to the conventional
Lindemann criterion, the found characteristic is explicitly expressed through
the graphene bending moduli of the second, third, and fourth orders.
