  Magnetic shape memory alloys are characterized by the coupling between a
structural phase transition and magnetic one. This permits to control the shape
change via an external magnetic field, at least in single crystals. Composite
materials with single-crystalline particles embedded in a softer matrix have
been proposed as a way to overcome the blocking of the transformation at grain
boundaries. We investigate hysteresis phenomena for small NiMnGa single
crystals embedded in a polymer matrix for slowly varying magnetic fields. The
evolution of the microstructure is studied within the rate-independent
variational framework proposed by Mielke and Theil (1999). The underlying
variational model incorporates linearized elasticity, micromagnetism, stray
field and a dissipation term proportional to the volume swept by the phase
boundary. The time discretization is based on an incremental minimization of
the sum of energy and dissipation. A backtracking approach is employed to
approximately ensure the global minimality condition. We illustrate and discuss
the influence of the particle geometry (volume fraction, shape, arrangement)
and the polymer elastic parameters on the observed hysteresis and compare with
recent experimental results.
