  We investigate phase-field modeling of brittle fracture in a one-dimensional
bar featuring a continuous variation of elastic and/or fracture properties
along its axis. Our main goal is to quantitatively assess how the heterogeneity
in elastic and fracture material properties influences the observed behavior of
the bar, as obtained from the phase-field modeling approach. The results
clarify how the elastic limit stress, the peak stress and the fracture
toughness of the heterogeneous bar relate to those of the reference homogeneous
bar, and what are the parameters affecting these relationships. Overall, the
effect of heterogeneity is shown to be strictly tied to the non-local nature of
the phase-field regularization. Finally, we show that this non-locality may
amend the ill-posedness of the sharp-crack problem in heterogeneous bars where
multiple points compete as fracture locations.
