One-dimensional localization studied with a second grade model

Second grade models are now used to include a meso scale in continuous models. Using the virtual power method, a simple one-dimensional second grade model based on the work of Germain, is presented. This model is a second grade generalization of a common softening model. Under the small strain assumption and simple assumptions about the external forces, analytical solutions of boundary value problems involving this model are established. They show that for such models non uniqueness can be proved. This implies that well-posedness is not automatically restored by adding a second grade term to a given first grade model. Then, a numerical analysis of boundary value problems, using this second grade model and the large strain assumption as well as the small strain one is developed. The corresponding one-dimensional finite element code is presented. The code is validated by comparison between analytical solutions and corresponding numerical ones, in the simple case of the small strain assumption. Using this validated numerical code, a comparison between small and large strain solutions, a study of mesh dependence and the influence of imperfections are also presented.