Soft robots that grow through eversion/apical extension can effectively navigate fragile environments such as ducts and vessels inside the human body. This paper presents the physics-based model of a miniature steerable eversion growing robot. We demonstrate the robot's growing, steering, stiffening and interaction capabilities. The interaction between two robot-internal components is explored, i.e., a steerable catheter for robot tip orientation, and a growing sheath for robot elongation/retraction. The behavior of the growing robot under different inner pressures and external tip forces is investigated. Simulations are carried out within the SOFA framework. Extensive experimentation with a physical robot setup demonstrates agreement with the simulations. The comparison demonstrates a mean absolute error of 10 - 20% between simulation and experimental results for curvature values, including catheter-only experiments, sheath-only experiments and full system experiments. To our knowledge, this is the first work to explore physics-based modelling of a tendon-driven steerable eversion growing robot. While our work is motivated by early breast cancer detection through mammary duct inspection and uses our MAMMOBOT robot prototype, our approach is general and relevant to similar growing robots.