Accuracy and numerical calculation time are the two main challenges of continuum robots dynamics modeling. In fact, the numerical calculation times of exact models are so long, that they are not practical in applications such as real-time control. This paper presents a new method for dynamics modeling of continuum robot backbones. In this method, the backbone shape is considered as an arbitrary number of constant-curvature (circular arc) elements, and the dynamics model is derived using Lagrange energy methods. First, kinetics and kinematics of one element are derived. Then, the robot kinematics is derived, as a series of such elements. Finally, the robot dynamics model is derived, using Euler-Lagrange method. This paper is focused on dynamics of the flexible body of continuum robots, and the proposed model is independent of actuation systems. Besides, the numerical singularity of the constant-curvature elements is avoided, which occurs when an element is straight. The model is validated using experimental results. Comparison of simulation and experimental results shows the accuracy of the proposed method on dynamics modeling. Furthermore, the calculation time of the model is short enough to make it practical for applications such as real-time control.