Explosive eruptions are the most powerful and destructive type of volcanic activity. These eruptions are characterized by magma fragmentation, the process through which a bubbly or foamy magma is transformed into a gas-pyroclast dispersion. Although magma fragmentation has been investigated both experimentally and theoretically, and the basic transport phenomena that occur in a volcanic conduit have been modelled, the underlying mechanism responsible for magma fragmentation is still poorly understood. This lack of understanding seriously limits our ability to forecast volcanic hazards, preventing reliable discrimination between conditions that lead to explosive and effusive eruptions. Here I develop a model in which a fragmentation criterion, based on a rate-limited crossing of the glass transition, , is incorporated into a multiphase fluid-dynamic description of magma ascent. The numerical results of this model demonstrate the feasibility of strain-induced brittle fragmentation of magma in volcanic eruptions, and reconcile experimental with theoretical studies as well as with the observed volcanic products of large explosive eruptions.