  We present an extension of fiber bundle models considering that failed fibers
still carry a fraction $0 \leq \alpha \leq 1$ of their failure load. The value
of $\alpha$ interpolates between the perfectly brittle failure $(\alpha = 0)$
and perfectly plastic behavior $(\alpha=1)$ of fibers. We show that the finite
load bearing capacity of broken fibers has a substantial effect on the failure
process of the bundle. In the case of global load sharing it is found that for
$\alpha \to 1$ the macroscopic response of the bundle becomes perfectly plastic
with a yield stress equal to the average fiber strength. On the microlevel, the
size distribution of avalanches has a crossover from a power law of exponent
$\approx 2.5$ to a faster exponential decay. For localized load sharing,
computer simulations revealed a sharp transition at a well defined value
$\alpha_c$ from a phase where macroscopic failure occurs due to localization as
a consequence of local stress enhancements, to another one where the disordered
fiber strength dominates the damage process. Analysing the microstructure of
damage, the transition proved to be analogous to percolation. At the critical
point $\alpha_c$, the spanning cluster of damage is found to be compact with a
fractal boundary. The distribution of bursts of fiber breakings shows a power
law behaviour with a universal exponent $\approx 1.5$ equal to the mean field
exponent of fiber bundles of critical strength distributions. The model can be
relevant to understand the shear failure of glued interfaces where failed
regions can still transmit load by remaining in contact.
