Analysis of non-local models through energetic formulations

Non-local formulations have been intensively applied for the past 20 years to control the strain localisation resulting from strain-softening constitutive laws. We propose a framework that encompasses most of these non-local approaches. It relies on an energetic interpretation of both equilibrium equations and constitutive relations for generalised standard materials. It allows a systematic exploration of the properties of these models. We focus here on three aspects: the existence of solutions when applied to strain-hardening laws, the effective localisation control when applied to strain-softening laws and the thermodynamic bases. A quite extended analysis is led under some assumptions, the more restrictive being that all the internal variables are dealt with the same regularising operator. It appears that only few approaches bring effective regularisation. Unfortunately, they do not rely on thermodynamic bases so that the Clausius–Duhem inequality is not automatically fulfilled. An alternative consists in giving up the assumptions of the analysis. In that case, it seems that no general result can be stated. Moreover, an illustration in the context of brittle damage shows that slightly different formulations can result in strongly differentiated behaviour, even though some specific choices lead nevertheless to a suitable model.