  Low mechanical loss materials are needed to further decrease thermal noise in
upcoming gravitational wave detectors. We present an analysis of the
contribution of Akhieser and thermoelastic damping on the experimental results
of resonant mechanical loss measurements. The combination of both processes
allows the fit of the experimental data of quartz in the low temperature region
(10 K to 25 K). A fully anisotropic numerical calculation over a wide
temperature range (10 K to 300 K) reveals, that thermoelastic damping is not a
dominant noise source in bulk silicon samples. The anisotropic numerical
calculation is sucessfully applied to the estimate of thermoelastic noise of an
advanced LIGO sized silicon test mass.
