This paper introduces a Cosserat rod based mathematical model for modeling a self-controllable variable curvature soft continuum robot. This soft continuum robot has a hollow inner channel and was developed with the ability to perform variable curvature utilizing a growing spine. The growing spine is able to grow and retract while modifies its stiffness through milli-size particle (glass bubble) granular jamming. This soft continuum robot can then perform continuous curvature variation, unlike previous approaches whose curvature variation is discrete and depends on the number of locking mechanisms or manual configurations. The robot poses an emergent modeling problem due to the variable stiffness growing spine which is addressed in this paper. We investigate the property of growing spine stiffness and incorporate it into the Cosserat rod model by implementing a combined stiffness approach. We conduct experiments with the soft continuum robot in various configurations and compared the results with our developed mathematical model. The results show that the mathematical model based on the adapted Cosserat rod matches the experimental results with only a 3.3\% error with respect to the length of the soft continuum robot.