Structured fluids (concentrated suspensions, emulsions, gels&cellip;.) typically exhibit an apparent yield stress. We show here that for a number of these fluids, a unique yield stress cannot be defined. Instead, when solicited above a critical stress, typical yield stress fluids (gels, clay suspensions) and soft glassy materials (colloidal glasses) start flowing abruptly and subsequently accelerate. We demonstrate that the competition between the spontaneous restructuration (aging) and the destruction of the internal structure (‘shear rejuvenation’) lead to a bifurcation in rheological behavior. For a stress smaller than a (time‐dependent) critical value, the viscosity increases in time and the material eventually stops flowing. For a slightly larger stresses the viscosity decreases continuously in time and the flow accelerates. Thus the viscosity jumps discontinuously to infinity at the critical stress. © 2004 American Institute of Physics