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M ′′ (ωP, T) = 3

M ′′ (ωP, T) = 3 DPl 2 πI (ΛSP − ΛP) (22) and internal friction Q −1 (ωP, T) were calculated Q −1 (ωP, T) = M ′′ (ωP, T) M ′ (ωP, T) = t2SP (ΛSP − ΛP) π(t2 P − t2 . (23) SP) The mechanical spectra are characterized by the complex Young’s modulus M ⋆ (T), complex shear modulus G ⋆ (T), internal friction Q −1 (T), complex creep compliance D ⋆ (T)M ⋆ (T) = 1, shear compliance J ⋆ (T)G ⋆ (T) = 1 and/or the complex shear viscosity η ⋆ (t). The complex quantities can be separated into real (storage modulus M ′ ) and imaginary parts (loss modulus M ′′ ). The Kramers-Kronig (dispersion) relations (Kramers (1926), Kronig (1926)) couple the real and imaginary parts of the complex quantities A ⋆ = A ′ + iA ′′ of a material by A ′ (ω) = −H[A ′′ (ξ)], A ′′ (ω) = H[A ′ (ξ)] (24) with the Hilbert-transform by principal value integral � (Donth (2001)) H[f(ξ)](ω) = 1 � f(ξ)dξ π ξ − ω , H−1 = −H. (25) These are purely mathematical implications of the linear and causal material equations. The discussion of the mechanical spectra can take place thereby on the individual modulus. 5 Results Fig.5 represents the storage modulus M ′ and the loss modulus M ′′ for the investigated natural glasses. All obsidians show a relatively similar behaviour. A deviation from the general trend shows the LIP sample due to strong vesiculation. Here, a particularly long-drawn-out thermal and mechanical glass transition can be observed in the storage modulus. The storage modulus at room temperature MRT is a material constant, i.e. the Young’s-modulus 2 of the glass E = MRT, which ranges for silicate glasses 2 If measurements of M ′ (T) are carried out below RT, one observes a further rise with falling temperature or rising frequency due to relaxation processes (Jagdt (1960)). 16

Fig. 5. Temperature dependence of the real part (storage modulus) M ′ and imaginary part (loss modulus) M ′′ of the complex Young’s modulus M ⋆ = M ′ + iM ′′ for the natural volcanic glasses. (inset) Asymptotic behaviour of storage modulus at hight temperature. 17

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