A coupled model for one-dimensional time-dependent compressible flow and bubble expansion is developed to investigate fragmentation mechanisms of viscous bubbly magmas in shock tubes. Initially a bubbly magma at a high pressure is separated from air at the atmospheric pressure by a diaphragm. As the diaphragm is ruptured, a shock wave propagates into the air, and a rarefaction wave propagates into the bubbly magma. As a result, the bubbly magma is decompressed and expands. Gas overpressure and hoop stress around expanding bubbles are calculated by applying the cell model. It is assumed that the magma fragments and the flow changes from bubbly flow to gas-pyroclast dispersion when the hoop stress or the gas volume fraction reaches a given threshold. Two types of fragmentation mechanisms are recognized: (1) high-viscosity magma fragments as the hoop stress reaches the tensile strength of the melt (stress fragmentation) and (2) the hoop stress does not grow in low-viscosity magma so that fragmentation occurs after bubble expansion when the gas volume fraction reaches a threshold (expansion fragmentation). During stress fragmentation a zone of steep pressure gradient forms just behind the fragmentation surface, which propagates into the magma together with the fragmentation surface. Analytical considerations suggest that the self-sustained stress fragmentation process can be described by a combination of a traveling-wave-type solution in the bubbly flow region and a self-similar solution in the gas-pyroclast flow region. Some simple formulae to predict the fragmentation speed (downward propagation velocity of the fragmentation surface) are derived on the basis of these solutions. The formulae are applied to recent experimental results using shock tube techniques as well as Vulcanian explosions in nature.