Control of robots with soft actuators is still challenging due to the complexity of modeling actuator material dynamics in conjunction with the robot dynamics. In this paper, we introduce a 45-mm-long soft continuum robot with distributed local actuators in the form of cubes composed of a novel temperature-responsive hydrogel, each with an embedded Joule heater. We refer to these as soft voxel actuators (SVAs). We present a dynamical model of this hydrogel-based continuum robot based on Cosserat rod theory. We experimentally identify the relationship between step input voltages applied to the SVAs and their resulting force outputs. In addition, we identify other unknown parameters of the model using vibration tests with the robot. We then numerically solve the Cosserat model and compare simulations of the model to measurements of the robot's tip displacement over time during open-loop control trials in which subsets of the SVAs are actuated. The normalized root-mean-square errors (NRMSEs) between the simulated and experimentally measured displacements are below 10%, which demonstrates the accuracy of the Cosserat model in describing the dynamics of the hydrogel-based continuum robot.