The Energy Balance of a Viscoelastic Material

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Abstract
It is well known that the work done by external forces on a viscoelastic material is converted into a conserved part (potential energy) and a dissipated part, each of which may be divided into two other parts: the isotropic one which is connected with volume changes and the deviatoric one which is associated with distortions. For strain and strain-rate-independent Poisson's ratio (which is reported to be the case for most viscoelastic materials) the time-dependent isotropic and deviatoric moduli differ only by a constant factor. Expressing the relaxation moduli by Prony-Dirichlet series enables the evaluation of the isotropic and deviatoric parts of the stress-power. These calculations are carried out for the case of constant strain-rate uniaxial tension. The positive definite terms of the resulting expression stand for the dissipated stress-power and the remaining terms—for the conserved stress-power. By integrating over time, the different parts of the stress-energy are obtained. The ratio of deviatoric part to isotropic part of energy is found to be independent of time and equal for both conserved and dissipated energies. Results of experiments carried out on Perspex (polymethyl methacrylate) and epoxy-resin were used to calculate the different parts of stress-energy. It is found that the ratio of dissipated energy to conserved energy is always smaller than unity decreasing for smaller strains and strain-rates. The energy computations are practically not affected by the choice of the parameters representing the viscoelastic behaviour of material. The proposed method can be easily applied to other experimental conditions such as relaxation, creep, constant rate of stress or any other loading history.

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