  We consider the quasi-static evolution of a brittle layer on a stiff
substrate; adhesion between layers is assumed to be elastic. Employing a
phase-field approach we obtain the quasi-static evolution as the limit of
time-discrete evolutions computed by an alternate minimization scheme. We study
the limit evolution, providing a qualitative discussion of its behaviour and a
rigorous characterization, in terms of parametrized balanced viscosity
evolutions. Further, we study the transition layer of the phase-field, in a
simplified setting, and show that it governs the spacing of cracks in the first
stages of the evolution. Numerical results show a good consistency with the
theoretical study and the local morphology of real life craquelure patterns.
