  We present a phase field-based framework for modelling fatigue damage in
Shape Memory Alloys (SMAs). The model combines, for the first time: (i) a
generalised phase field description of fracture, incorporating multiple phase
field formulations, (ii) a constitutive model for SMAs, based on a
Drucker-Prager form of the transformation surface, and (iii) a fatigue
degradation function, with damage driven by both elastic and transformation
strains. The theoretical framework is numerically implemented, and the
resulting linearised system is solved using a robust monolithic scheme, based
on quasi-Newton methods. Several paradigmatic boundary value problems are
addressed to gain insight into the role of transformation stresses,
stress-strain hysteresis and temperature. Namely, we compute $\Delta
\varepsilon-N$ curves, quantify Paris law parameters and predict fatigue crack
growth rates in several geometries. In addition, the potential of the model for
solving large-scale problems is demonstrated by simulating the fatigue failure
of a 3D lattice structure.
