  Colloidal gels consist of percolating networks of interconnected arms. Their
mechanical properties depend on the individual arms, and on their collective
behaviour. We use numerical simulations to pull on a single arm, built from a
model colloidal gel-former with short-ranged attractive interactions. Under
elongation, the arm breaks by a necking instability. We analyse this behaviour
at three different length scales: a rheological continuum model of the whole
arm; a microscopic analysis of the particle structure and dynamics; and the
local stress tensor. Combining these different measurements gives a coherent
picture of the necking and failure: the neck is characterised by plastic flow
that occurs for stresses close to the arm's yield stress. The arm has an
amorphous local structure and has large residual stresses from its
initialisation. We find that neck formation is associated with increased
plastic flow, a reduction in the stability of the local structure, and a
reduction in the residual stresses; this indicates that {the} system loses its
solid character and starts to behave more like a viscous fluid. We discuss the
implications of these results for the modelling of gel dynamics.
