The tip asymmetry of a bevel-tip needle results in the needle naturally bending when it is inserted into soft tissue. This enables robotic needle steering, which can be used in medical procedures to reach subsurface targets inaccessible by straight-line trajectories. However, accurate path planning and control of needle steering requires models of needle-tissue interaction. Previous kinematic models required empirical observations of each needle and tissue combination in order to fit model parameters. This study describes a mechanics-based model of robotic needle steering, which can be used to predict needle behavior and optimize system design based on fundamental mechanical and geometrical properties of the needle and tissue. We first present an analytical model for the loads developed at the tip, based on the geometry of the bevel edge and material properties of soft-tissue simulants (gels). We then present a mechanics-based model that calculates the deflection of a bevel-tipped needle inserted through a soft elastic medium. The model design is guided by microscopic observations of needle-gel interactions. The energy-based formulation incorporates tissue-specific parameters, and the geometry and material properties of the needle. Simulation results follow similar trends (deflection and radius of curvature) to those observed in experimental studies of robotic needle insertion.