  We consider fragmentation processes with values in the space of marked
partitions of $\mathbb{N}$, i.e. partitions where each block is decorated with
a nonnegative real number. Assuming that the marks on distinct blocks evolve as
independent positive self-similar Markov processes and determine the speed at
which their blocks fragment, we get a natural generalization of the
self-similar fragmentations of Bertoin (2002). Our main result is the
characterization of these generalized fragmentation processes: a
L\'evy-Khinchin representation is obtained, using techniques from positive
self-similar Markov processes and from classical fragmentation processes. We
then give sufficient conditions for their absorption in finite time to a frozen
state, and for the genealogical tree of the process to have finite total
length.
