  Phase mixing of Alfv\'en waves has been studied extensively as a possible
coronal heating mechanism but without the full thermodynamic consequences
considered self-consistently. It has been argued that in some cases, the
thermodynamic feedback of the heating could substantially affect the transverse
density gradient and even inhibit the phase mixing process. In this paper, we
use MHD simulations with the appropriate thermodynamical terms included to
quantify the evaporation following heating by phase mixing of Alfv\'en waves in
a coronal loop and the effect of this evaporation on the transverse density
profile. The numerical simulations were performed using the Lare2D code. We set
up a 2D loop model consisting of a field-aligned thermodynamic equilibrium and
a cross-field (background) heating profile. A continuous, sinusoidal,
high-frequency Alfv\'en wave driver was implemented. As the Alfv\'en waves
propagate along the field, they undergo phase mixing due to the cross-field
density gradient in the coronal part of the loop. We investigated the presence
of field-aligned flows, heating from the dissipation of the phase-mixed
Alfv\'en waves, and the subsequent evaporation from the lower atmosphere. We
find that phase mixing of Alfv\'en waves leads to modest heating in the shell
regions of the loop and evaporation of chromospheric material into the corona
with upflows of the order of only 5-20 m/s. Although the evaporation leads to a
mass increase in the shell regions of the loop, the effect on the density
gradient and, hence, on the phase mixing process, is insignificant. This paper
self-consistently investigates the effect of chromospheric evaporation on the
cross-field density gradient and the phase mixing process in a coronal loop. We
found that the effects in our particular setup (small amplitude, high frequency
waves) are too small to significantly change the density gradient.
