In percutaneous needle insertions, cutting forces at the needle tip deflect the needle and increases targeting error. Thus, modeling needle-tissue interaction in biological tissue is essential for accurate robotics-assisted needle steering. In this letter, dynamics of needle tip interaction with inhomogeneous biological tissue is described and the effects of insertion velocity, tissue mechanical characteristics, and needle geometry on tissue cutting force are studied. Needle interaction with biological tissue is divided into three distinct events and modeled. 1) Initial tissue puncturing, which starts by soft tissue deformation and continues until a crack is formed in the tissue. Employing a viscoelastic model of fracture initiation we have predicted the maximum puncturing force and force-displacement response of a needle in contact with a tissue. 2) Tissue cutting, which follows the crack propagation in tissue and is predicted using a novel energy-based fracture model. The model takes account of the needle tip geometry and the tissue mechanical characteristics. 3) Friction between tissue and needle shaft is estimated during needle insertion and retraction using a needle-tissue friction model. Using a needle driving robot ex vivo experiments are performed on a porcine tissue sample to identify the model parameters and validate the analytical predictions offered by the models.