Inspired by the embodied intelligence of biological creatures like the octopus, a soft robotic arm uses its highly flexible structure to perform various tasks in a complex environment. While the classic Cosserat rod theory investigates the bending, twisting, shearing and stretching of the soft arm, it fails to capture the in-plane deformation that occurs during certain tasks, particularly those involving active lateral traction. This paper introduces an extended Cosserat rod theory addressing these limitations by incorporating an extra strain variable, reflecting the in-plane inflation ratio. To accurately describe the viscoelasticity effect of the soft body in dynamics, the proposed model enhances the constitutive law by integrating the Saint–Venant Kirchhoff (SVK) hyperelastic and Kelvin–Voigt viscous models. The active and environmental loads are accounted for by the equations of motion, which are numerically solved by adapting the geometric variable strain (GVS) approach to balance the accuracy and computational efficiency. Our contributions include the derivation of the extended Cosserat rod theory in a dynamic context, and the development of a reduced-order numerical method that enables rapid and precise solutions. We demonstrate applications of the model in the stiffness tuning of a soft robotic arm and the study of complex octopus arm motions.