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WagnerJVGR05.pdf

WagnerJVGR05.pdf

Since 0 ≤ β < α ≤

Since 0 ≤ β < α ≤ 1 the condition for thermodynamic compatibility is fulfilled (Heymans and Bauwens (1994), Heymans (1996)). Transformation into a standard model is carried out by adding an elastic element of modulus M1 in series. The dynamic response M ⋆ (ω, τ) and internal friction Q −1 then become: M⋆ (ω, τ) = M0 Q −1 (ω, τ) = M1(iωτ) α M1(1 + (iωτ) α−β ) + M0(iωτ) α, D ⋆ (ω, τ) − D1 D0 = (iωτ) −α + (iωτ) −β (ωτ) −α sin � � απ + (ωτ) 2 −β sin � � βπ 2 (ωτ) −α cos � � απ + (ωτ) 2 −β cos � � βπ 2 + M0 M1 (17) (18) It is also possible to give an analytical expression for the relaxation time distribution h(τ) with the relation M ⋆ (ω, τ) = ˆ M(p, τ)|p=iω Gloeckle (1993): h(τ) = ± lim ε→0 ℑ[ ˆ M(p)]|p=−1/τ±ε and one gets: h(τ) = Ω S α,β + M0 M1 ˜τ −2α + ˜τ −2β + 2˜τ −α−β cos([α − β]π) + 2M0 M1 ΩC α,β + M2 0 M 2 1 (19) (20) with ˜τ = τ/τmax, Ω C i,j = ˜τ −i cos(iπ) + ˜τ −j cos(jπ), Ω S i,j = ˜τ −i sin(iπ) + ˜τ −j sin(jπ). 12

13 Table 2 The summary of various spectral functions and their power-law behaviour. Abbreviation: p = iωτ, A ′ ,A - constants. ωτ → 0 ωτ → ∞ M ⋆ /M0 M ′ /M0 M ′′ /M0 Q −1 M ′ /M0 M ′′ /M0 Q −1 Debye (1941) (1 + p −1 ) −1 (ωτ) 2 (ωτ) 1 (ωτ) −1 1 (ωτ) −1 (ωτ) −1 Cole and Cole (1941) (1 + p −α ) −1 0 < α ≤ 1 (ωτ) α (ωτ) α A ′ 1 (ωτ) −α (ωτ) −α Jonscher (1977) (p −α + p −β ) −1 0 < β ≤ α ≤ 1 (ωτ) α (ωτ) α A ′ (ωτ) β (ωτ) β A KWW (Kohlrausch (1847)) exp(−(t/τ) β ) 0 < β ≤ 1 (ωτ) 2 (ωτ) 1 (ωτ) −1 1 (ωτ) −β (ωτ) −β Havriliak and Negami (1967) 1 − (1 + p α ) −β 0 < α,β < 1 (ωτ) α (ωτ) α A ′ 1 (ωτ) −βα (ωτ) −βα Davidson and Cole (1951) 1 − (1 + p) −β 0 < β ≤ 1 (ωτ) 2 (ωτ) 1 (ωτ) −1 1 (ωτ) −β (ωτ) −β Mueller (1983) Q = (ωτ) γ 0 < γ ≤ 1 → 0 (ωτ) 1 (ωτ) −γ 1 (ωτ) −γ (ωτ) −γ FC1 (p −1 + p −β + M0 M1 )−1 0 < β ≤ 1 (ωτ) 2−β (ωτ) 1 (ωτ) −1+β 1 (ωτ) −β (ωτ) −β FC2 (p −α + p −β + M0 M1 )−1 0 < β ≤ α ≤ 1 (ωτ) α (ωτ) α A ′ 1 (ωτ) −β (ωτ) −β

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