Heterogeneous materials can be seen as bulk-interface systems. They
consist of distinct bulk components with different material properties
meeting at thin interfacial layers forming lower-dimensional substructures
of the system. In many applications the properties of interfaces strongly
impact the functionality of the whole system and, in turn,
interfaces are strongly affected by processes taking place in the bulk
material. Interfaces thus follow their own evolution laws in interaction
with bulk processes.

In this talk I discuss a general thermodynamical modeling framework for
bulk-interface processes and, in particular, apply it to problems related
to heat conduction and fracture in elastic composites. Here, a challenge
in the modeling and in the analysis lies in the change of the material
geometry with the progressing fracture and in the constraint that in many
materials crack growth is a unidirectional process, since the crack cannot
heal.

Models suited to handle these challenges and thus suited to describe
dynamic fracture processes in elastic solids with the aid of non-smooth
constraints will be introduced. Recent results on their mathematical
analysis will be presented.