The interpretation of dynamic processes that occur in volcanic calderas is not simple. The ground deformations and the local seismicity, which in other volcanic contexts are usually regarded as precursors to eruption, in caldera environment, in many cases, are not followed by any eruption. We formulate a general hypothesis that can explain these behaviours. Our hypothesis is that the intrusion of a sill can be responsible for the dynamics observed during unrest at calderas. In order to investigate the reliability of this hypothesis we developed a dynamic model of sill intrusion in a shallow volcanic environment. In our model the sill, fed by a deeper magma reservoir, intrudes below a horizontal elastic plate, representing the overlying rocks, and expands with axis-symmetric geometry. The model is based on the numerical solution of the equation for the elastic plate, coupled with a Navier-Stokes equation for simulating the dynamics of the sill intrusion. We performed a number of simulations, with the objective of showing the main features of the model. In the experiments, when the feeding process stops, the vertical movement reverses its trend and the area of maximum uplift undergoes subsidence. Under certain conditions the subsidence can occur even during the intrusion of the sill. The stress-field produced by the intrusion is mainly concentrated in a circular zone that follows the sill intrusion front. The features predicted by the model are consistent with many observations carried out on different calderas as reported in the scientific literature.