  State of the art nanomechanical resonators present quality factors Q ~ 10^3 -
10^5, which are much lower than those that can be naively extrapolated from the
behavior of micromechanical resonators. We analyze the dissipation mechanism
that arises in nanomechanical beam-structures due to the tunneling of
mesoscopic phonons between the beam and its supports (known as clamping
losses). We derive the environmental force spectral density that determines the
quantum Brownian motion of a given resonance. Our treatment is valid for low
frequencies and provides the leading contribution in the aspect ratio. This
yields fundamental limits for the Q-values which are described by simple
scaling laws and are relevant for state of the art experimental structures. In
this context, for resonant frequencies in the 0.1-1GHz range, while this
dissipation mechanism can limit flexural resonators it is found to be
negligible for torsional ones. In the case of structureless 3D supports the
corresponding environmental spectral densities are Ohmic for flexural
resonators and super-Ohmic for torsional ones, while for 2D slab supports they
yield 1/f noise. Furthermore analogous results are established for the case of
suspended semiconducting single-walled carbon nanotubes. Finally, we provide a
general expression for the spectral density that allows to extend our treatment
to other geometries and illustrate its use by applying it to a microtoroid. Our
analysis is relevant for applications in high precision measurements and for
the prospects of probing quantum effects in a macroscopic mechanical degree of
freedom.
