Gradient Damage Models In Large Deformation

In 1998, Francfort and Marigo proposed a variational approach to fracture (Francfort & Marigo 1998) in which the whole damage or crack evolution is governed by a least energy principle. In the framework of brittle materials, with softening damage behaviour, they constructed damage gradient models which rely on three principles: irreversibility of the damage evolution, stability and energy balance. They introduced regularized damage models: the total energy contains terms with spatial damage gradient weighted by a parameter called "characteristic length". These models were thoroughly developed in the past years (Marigo, Maurini, & Pham 2016) and enriched with couplings with plasticity (Alessi, Marigo, & Vidoli 2014), dynamics, temperature (Sicsic & Marigo 2013). Yet they stayed in the framework of small strains. The aim of this work is therefore to study the relevance of theses models in finite strains. To this end, both an analytical and a numerical study were conducted. The homogeneous and localized analytical response of a damaging hyperelastic 1D bar submitted to a quasi-static traction test were first established, then compared to the ones obtained via a 1D numerical implementation using the FEniCS library. Once the 1D analytical and numerical study has been conducted, numerical simulations were performed in higher dimensions with the finite element academic code MEF++. Compressible or quasi-incompressible neo-Hookean and Mooney-Rivlin solid laws were used on different geometries.