Soft robots differ fundamentally from traditional robotics. Due to their composition of soft materials, soft robots are inherently compliant and allow for large continuum-bodied motion. Although some frameworks exist for describing the kinematics, the development of dynamic models intended for control-oriented applications is relatively scarce and, to some extent, underdeveloped. This paper provides a modeling framework to describe the nonlinear dynamics of a soft robot using differential geometry of spatial curves. Furthermore, we include the geometrically nonlinear and time-variant mechanical nature imposed by these soft materials into our modeling framework. Numerical simulations of the dynamic model are presented, as well as experimental validations of a study case soft robot to illustrate the accuracy. The proposed modeling framework can be used to simulate the nonlinear dynamics of soft robots but also to calculate the inverse dynamics required for model-based control.