Table 3 Simple expressions for estimating fundamental material losses using Eq. ( 2 )[5, 6]

Thermoelastic damping (TED)

${\Delta }_{\mathrm{TED}}=\frac{\mathit{E}{\mathit{\alpha }}^2\mathit{T}}{\mathit{C}}$

${\mathit{\tau }}_{\mathrm{TED}}=\frac{\mathit{C}{\mathit{h}}^2}{{\mathit{\pi }}^2\mathit{k}}$

Akhiezer damping

${\Delta }_{\mathrm{Akhiezer}}=\frac{\mathit{C}\mathit{T}{\mathit{\gamma }}^2}{\mathit{\rho }{\mathrm{v}}^2}$

${\mathit{\tau }}_{\mathrm{Akhiezer}}=\frac{3\mathit{k}}{\mathit{C}{\mathrm{v}}^2}$