During Drosophila gastrulation, the ventral mesodermal cells constrict their apices, undergo a series of coordinated cell-shape changes to form a ventral furrow (VF) and are subsequently internalized. Although it has been well documented that apical constriction is necessary for VF formation, the mechanism by which apical constriction transmits forces throughout the bulk tissue of the cell remains poorly understood. In this work, we develop a computational vertex model to investigate the role of the passive mechanical properties of the cellular blastoderm during gastrulation. We introduce to our knowledge novel data that confirm that the volume of apically constricting cells is conserved throughout the entire course of invagination. We show that maintenance of this constant volume is sufficient to generate invagination as a passive response to apical constriction when it is combined with region-specific elasticities in the membranes surrounding individual cells. We find that the specific sequence of cell-shape changes during VF formation is critically controlled by the stiffness of the lateral and basal membrane surfaces. In particular, our model demonstrates that a transition in basal rigidity is sufficient to drive VF formation along the same sequence of cell-shape change that we observed in the actual embryo, with no active force generation required other than apical constriction.