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We use numerical simulations and an athermal quasistatic shear protocol to investigate the yielding of a model colloidal gel. Under increasing deformation, the elastic regime is followed by a significant stiffening before yielding takes place. A space-resolved analysis of deformations and stresses unravel how the complex load curve observed is the result of stress localization and that the yielding can take place by breaking a very small fraction of the network connections. The stiffening corresponds to the stretching of the network chains, unbent, and aligned along the direction of maximum extension. It is characterized by a strong localization of tensile stresses that triggers the breaking of a few network nodes at around 30% of strain. Increasing deformation favors further breaking but also shear-induced bonding, eventually leading to a large-scale reorganization of the gel structure at the yielding. At low enough shear rates, density and velocity profiles display significant spatial inhomogeneity during yielding in agreement with experimental observations.


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