The breakup of small solid aggregates in homogeneous and isotropic turbulence is studied theoretically and by using direct numerical simulations at high Reynolds number, Reλ =/~ 400. We show that turbulent fluctuations of the hydrodynamic stress along the aggregate trajectory play a key role in determining the aggregate mass distribution function. The differences between turbulent and laminar flows are discussed. A definition of the fragmentation rate is proposed in terms of the typical frequency at which the hydrodynamic stress becomes sufficiently high to cause breakup along each Lagrangian path. We also define an Eulerian proxy of the real fragmentation rate, based on the joint statistics of the stress and its time derivative, which should be easier to measure in any experimental setup. Both our Eulerian and Lagrangian formulations define a clear procedure for the computation of the mass distribution function due to fragmentation. Contrary, previous estimates based only on single point statistics of the hydrodynamic stress exhibit some deficiencies. These are discussed by investigating the evolution of an ensemble of aggregates undergoing breakup and aggregation.