  We have measured the thermal properties of suspended membranes from 10 K to
300 K for two amplitudes of internal stress (about 0.1 GPa and 1 GPa) and for
two different thicknesses (50 nm and 100 nm). The use of the original 3 \omega
-Volklein method has allowed the extraction of both the specific heat and the
thermal conductivity of each SiN membrane over a wide temperature range. The
mechanical properties of the same substrates have been measured at helium
temperatures using nanomechanical techniques. Our measurements show that the
thermal transport in freestanding SiN membranes is not affected by the presence
of internal stress. Consistently, mechanical dissipation is also unaffected
even though Qs increase with increasing tensile stress. We thus demonstrate
that the theory developed by Wu and Yu [Phys. Rev. B 84, 174109 (2011)] does
not apply to this amorphous material in this stress range. On the other hand,
our results can be viewed as a natural consequence of the "dissipation
dilution" argument [Y. L. Huang and P. R. Saulson, Rev. Sci. Instrum. 69, 544
(1998)] which has been introduced in the context of mechanical damping.
