ONE-DIMENSIONAL LOCALIZATION STUDIED WITH A 2ND 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 seco nd grade model based on the work of Germain, is presented. This model is a second grade generalisation of a common softening model. Under th e small strain assumption and simple assumptions about the external fo rces, analytical solutions of boundary value problems involving this m odel are established. They show that for such models non uniqueness ca n be proved. This implies that well-posedness is not automatically res tored by adding a second grade term to a given first grade model. Then , a numerical analysis of boundary value problems, using this second g rade model and the large strain assumption as well as the small strain one is developed. The corresponding one-dimensional finite element co de is presented. The code is validated by comparison between analytica l solutions and corresponding numerical ones, in the simple case of th e small strain assumption. Using this validated numerical code, a comp arison between small and large strain solutions, a study of mesh depen dence and the influence of imperfections are also presented. (C) Elsev ier, Paris.