Abstract

Gradient-enhanced models based on regularised displacements are an alternative to standard models in order to simulate material failure. In this alternative formulation, mechanical displacements u coexist with smoothed displacements u˜, which are the solution of a diffusion-reaction equation. Analogously to regularised standard models, prescribing boundary conditions in this alternative formulation is an open problem. Nevertheless, imposing these conditions for the displacement field (rather than the internal state variable) seems to be easier to interpret. The goal of this work is to study the influence of these conditions: neither Dirichlet conditions (prescribed at the beginning) nor homogeneous Neumann conditions (reminiscence of standard gradient models) allow to obtain realistic numerical results, on the other hand, by means of non-homogeneous Neumann boundary conditions, these results are physically admissible. However, these conditions do not ensure volume conservation, which is an interesting property in some constitutive models. Hence, new conditions (combined conditions) are proposed. These satisfy the necessary properties for regularisation: (a) reproducibility of order 1 (u=u˜ if u is a linear field), (b) displacement smoothing along the boundary and (c) volume preservation. In this work various two-dimensional numerical tests are carried out in order to illustrate the influence of the different boundary conditions.