Relevance of symmetry methods in mechanics of materials

The interest and relevance of symmetry methods as a predictive and systematic methodology in the continuum mechanics of materials is analyzed, relying on a classification of the inherent aspects in terms of the direct, extended direct, and inverse methods. Although being interrelated, these three problems each have a specific argumentation which is separately exposed in the present contribution. The direct problem of finding invariants associated with a given constitutive law for materials, including dissipation, is first envisaged. The abstract formulation of constitutive laws in terms of the state laws and a dissipation potential expressing the evolution of internal state variables is considered, in the framework of irreversible thermodynamics. It is shown that a specific choice of the components of the symmetry vector acting in the space of independent and dependent variables leads to a local invariance condition of the constitutive law fully equivalent to the variational symmetry condition using the rate of the internal energy density. As a specific situation involving this methodology, a time-temperature equivalence principle of polymers is obtained from the requirement of group invariance of the field equations. A validation of this invariance principle is given by a comparison of the modelled master response and the master curve constructed from a set of experimental results at various temperatures. The extended direct method is next presented as a generalization of the direct method, in the sense that a classification of constitutive functions modelling the material behavior is achieved via a symmetry analysis. In the third part of the paper, the inverse problem of constructing a material's constitutive law exploiting a postulated Lie-group structure is exposed. A constitutive model is then identified which satisfies the symmetries exhibited by the experimental data.