  We study theoretically the dynamical process of yielding in cyclically
sheared amorphous materials, within a thermal elastoplastic model and the soft
glassy rheology model. Within both models we find an initially slow
accumulation, over many cycles after the inception of shear, of low levels of
damage in the form strain heterogeneity across the sample. This slow fatigue
then suddenly gives way to catastrophic yielding and material failure. Strong
strain localisation in the form of shear banding is key to the failure
mechanism. We characterise in detail the dependence of the number of cycles N*
before failure on the amplitude of imposed strain, the working temperature, and
the degree to which the sample is annealed prior to shear. We discuss our
finding with reference to existing experiments and particle simulations, and
suggest new ones to test our predictions.
