Gradient constitutive relations: numerical aspects and application to gradient damage

The description of localisation and structural failure through local strain-softening constitutive laws leads to ill-posed problems. To retrieve a well-posed problem, the material state is described through the gradients of internal variables in addition to the strain and the internal variables themselves. Under some assumptions, it is shown that these gradient laws can be cast into a variational formulation. After time discretisation, the evolution of the internal variable field is characterised by a minimum principle. A specific algorithm is proposed to perform the minimisation in the context of finite elements. Finally, it is applied to a gradient brittle damage model.