Display Abstract

Title Non-smooth degenerating elliptic equations for damage models

Name Elena Bonetti
Country Italy
Email elena.bonetti@unipv.it
Co-Author(s) Francesco Freddi (Univ. Parma), Antonio Segatti (Univ. Pavia)
Submit Time 2014-02-27 07:04:47
Special Session 2: Nonlinear evolution PDEs and interfaces in applied sciences
The thermo-mechanical behaviour of (visco)elastic and special materials is determined by the coexistence of different configurations at the micro-scale, changing due to thermo-mechanical loads (as in the case of degrading mechanical properties in quasi-brittle materials). The continuum damage theory, introduced in the framework of solid-solid phase transitions, turns out to perform a good description of the evolving damage phenomenon in materials losing their stiffness. In particular, an order parameter delineates the state of the cohesion in the body. The resulting PDE system we investigate couples thermo-mechanical actions and phase dynamics and it is strongly nonlinear and possibly degenerating in the elliptic equations for deformations. We discuss existence of a solution in the case of complete damage. This problem is still open in its original formulation, but some answer has been given for suitable weak notion of solutions. We introduce a new notion of solution, accounting for a new "interior" stress which implicitly accounts for the state of damage of the material and write an appropriate free energy. We recover a new PDE system describing complete damage and we show the existence of a weak solution.