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On the Lie Symmetry Groups With Application to a Non Linear Viscoelastic Behaviour

The continuous symmetry groups of a viscoelastic constitutive behaviour are calculated. The considered constitutive law is expressed within a thermodynamic approach of irreversible processes, called the Distribution of Non-linear Relaxations (acronym DNLR). Interpreting the material behaviour as a partial differential equation satisfied by the stress as a function of the time and of a set of continuously varying parameters, its continuous symmetries are determined, from the generator vector fields associated to each infinitesimal symmetry. The expression of the symmetry groups makes it possible to find a master curve and to underline the well-known time temperature equivalence principle. The symmetry analysis provides a rational and systematic way of construction of the master curves, that appears more economical in terms of experiments compared to the traditional heuristic approach.