  This paper studies the competition of fracture initiation in the ductile soft
phase and in the comparatively brittle hard phase in the microstructure of a
two-phase material. A simple microstructural model is used to predict
macroscopic fracture initiation. The simplicity of the model ensures highly
efficient computations, enabling an comprehensive study: a large range of hard
phase volume fractions and yield stress ratios, for wide range of applied
stress states. Each combination of these parameters is analyzed using a large
set of (random) microstructures. It is observed that only one of the phases
dominates macroscopic fracture initiation: at low stress triaxiality the soft
phase is dominant, but above a critical triaxiality the hard phase takes over
resulting in a strong decrease in ductility. This transition is strongly
dependent on microstructural parameters. If the hard phase volume fraction is
small, the fracture initiation is dominated by the soft phase even at high
phase contrast. At higher hard phase volume fraction, the hard phase dominates
already at low phase contrast. This simple model thereby reconciles
experimental observations from the literature for a specific combination of
parameters, which may have triggered contradictory statements in the past. A
microscopic analysis reveals that the average phase distribution around
fracture initiation sites is nearly the same for the two failure mechanisms.
Along the tensile direction, regions of the hard phase are found directly next
to the fracture initiation site. This `band' of hard phase is intersected
through the fracture initiation site by `bands' of the soft phase aligned with
shear. Clearly, the local mechanical incompatibility is dominant for the
initiation of fracture, regardless whether fracture initiates in the soft or in
the hard phase.
