Augmented Hooke's law based on alternative stress relaxation models

Relationships are presented which are considered to be the most general, linear, constitutive stress-strain relationships for a simple, solid material at isothermal conditions. Formally, they are obtained in frequency domain by adding "anelastic", frequency dependent terms to the elastic constants of the material. The resulting, basic, augmented Hooke's law (AHL) is proposed as a general framework for comparison of alternative linear material damping models. Contained, as special cases, are both the classical, purely mechanical theory of viscoelasticity and more recent damping models based on linear, irreversible thermodynamics. The original AHL Helmholtz free energy density function, Dovstam (1995), is generalised to materials with continuously distributed relaxation frequency spectra. It is shown that there corresponds a continuously distributed relaxation spectrum to each admissible linear damping model and how such relaxation spectra may be computed using the Stieltjes-Perron inversion formula and explicit (analytical) models of the complex, frequency dependent, parts of a corresponding AHL. Traditional relaxation time spectra (discrete or continuously distributed relaxation times) are shown to be directly related to AHL relaxation amplitude distributions (relaxation frequency spectral derived in the paper. The relationships between traditional relaxation time spectra and frequency domain AHL relaxation amplitude distributions connect experimental time domain data in linear viscoelasticity, with corresponding AHL relaxation frequency spectra which may be used in linear, constitutive material damping modelling. The results indicate that the information supplied by relaxation spectra tin time or frequency domain) is completely equivalent to any suitable and physically realistic damping model, properly curve fitted to experimental complex material moduli. Fractional derivative models are demonstrated to simulate the mean properties of the relaxation processes in the material during vibration. In this context, fractional derivative models are completely equivalent to frequency domain, continuously distributed AHL relaxation models, with well defined and easily computed relaxation frequency spectra. Using experimentally estimated data, it is explicitly demonstrated that linear, material damping may be simulated using discrete as well as continuously distributed AHL relaxation models or corresponding fractional derivative models. Which damping model to use is a matter of convenience.