  Thermal properties of graphene display peculiar characteristics associated to
the two-dimensional nature of this crystalline membrane. These properties can
be changed and tuned in the presence of applied stresses, both tensile and
compressive. Here we study graphene monolayers under tensile stress by using
path-integral molecular dynamics (PIMD) simulations, which allows one to take
into account quantization of vibrational modes and analyze the effect of
anharmonicity on physical observables. The influence of the elastic energy due
to strain in the crystalline membrane is studied for increasing tensile stress
and for rising temperature (thermal expansion). We analyze the internal energy,
enthalpy, and specific heat of graphene, and compare the results obtained from
PIMD simulations with those given by a harmonic approximation for the
vibrational modes. This approximation turns out to be precise at low
temperatures, and deteriorates as temperature and pressure are increased. At
low temperature the specific heat changes as $c_p \sim T$ for stress-free
graphene, and evolves to a dependence $c_p \sim T^2$ as the tensile stress is
increased. Structural and thermodynamic properties display nonnegligible
quantum effects, even at temperatures higher than 300~K. Moreover, differences
in the behavior of the in-plane and real areas of graphene are discussed, along
with their associated properties. These differences show up clearly in the
corresponding compressibility and thermal expansion coefficient.
