A singular limit of a family of variational evolutions for a brittle elastic bi-layer

We consider a system of one-dimensional elastic and brittle media coupled by an elastic term. Both the coupling and the fracture-energy coefficient depend on a small parameter
ɛ
. We describe the quasistatic motion of this system under varying boundary conditions at
ɛ
fixed, interpreting the fracture energy term as a dissipation as in the variational approach to fracture growth, by analysing a time-discrete approximation. The motion is characterized by the onset of successive fracture sites. We study the behaviour of these evolutions with vanishing dissipations as
ɛ
, characterizing the possible limit of the energies by a one-parameter family of time-parameterized functions. We show that, after an initial time interval, a function in this family corresponds to a minimum of the
-limit of the energies of the system only for a discrete set of the values of the quantity parameterizing boundary conditions.