Gilles Francfort L.P.M.T.M., Universit\'e Paris-Nord
Paris, France

A Variational Approach to Quasi-Static Evolution for a Class of Dissipative Materials

Abstract: Rate independence is a shared feature of many constitutive behaviors for solids, from brittle fracture, to associated elasto-plasticity, damage or phase transformation.

I will first discuss why any quasi-static evolution for rate-independent materials can be viewed as a time-indexed sequence of constrained minimization problems.

I will illustrate this in various settings chosen among fracture, phase transformations or damage.

The most natural mathematical approach consists in discretizing time and minimizing the corresponding energy. It then remains to pass to the limit in the time step.

This program is met with success in many cases like fracture or phase transformation.

I will indicate the main steps in the case of phase transformation for hyperelastic materials.

Then, I will focus on the setting of damage in the simplistic framework of a material that can only experience two states, a healthy state and a damaged state. The model quickly leads to relaxation because the material wants to form fine mixtures of its two available states, so as to minimize its total energy among all admissible material configurations. But, during the evolution, this process is opposed by the irreversible character of the damage process, because the latter prohibits to undo already formed microstructures. This last hurdle frustrated any further analysis for quite a while.

I will show how to solve the dichotomy and indicate how one can produce a relaxed evolution that preserves irreversibility and energy conservation while attaining minimality of the material configuration at each time. In contrast with the settings of fracture, elasto-plasticity or phase transformation, it is not so clear that the obtained evolution is that which is best suited to the damage process. The interaction between evolution and formation of microstructures remains a delicate topic, especially when irreversibility is present.