Initiation and propagation of fracture in the models of Griffith and Barenblatt

In the setting of the simplest debonding problem we give a systematic comparison of the fracture models due to Griffith and Barenblatt. We prove that the Griffith model represents an asymptotic -limit of the Barenblatt model, when the ratio of the external and internal lengths increases indefinitely. We then illustrate the character of convergence by solving explicitly two sample problems with initially rigid and initially elastic cohesive energies. The geometrical simplicity of the setting allows us to study the small parameter dependence of both global and local minimizers of the total energy.Received: 22 April 2003, Accepted: 12 September 2003, Published online: 9 January 2004PACS: 62.20.Mk, 68.35.Md, 81.40.Jj, 83.50.LhCorrespondence to: J.-J. Marigo     

The issues of the uniqueness and the stability of the homogeneous response in uniaxial tests with gradient damage models

We consider a wide class of gradient damage models which are characterized by two constitutive functions after a normalization of the scalar damage parameter. The evolution problem is formulated following a variational approach based on the principles of irreversibility, stability and energy balance. Applied to a monotonically increasing traction test of a one-dimensional bar, we consider the homogeneous response where both the strain and the damage fields are uniform in space. In the case of a softening behavior, we show that the homogeneous state of the bar at a given time is stable provided that the length of the bar is less than a state dependent critical value and unstable otherwise. However, we also show that bifurcations can appear even if the homogeneous state is stable. All these results are obtained in a closed form. Finally, we propose a practical method to identify the two constitutive functions. This method is based on the measure of the homogeneous response in a situation where this response is stable without possibility of bifurcation, and on a procedure which gives the opportunity to detect its loss of stability. All the theoretical analyses are illustrated by examples.