This paper introduces an interactive model of needle insertion, including the possibility to simulate lacerations of tissue around the needle. The method relies on complementary constraints to couple the Finite Element models of the needle and tissue. The cutting path is generated from mechanical criteria (i.e. cutting force) at arbitrary resolution, avoiding expensive remeshing of Finite Element meshes. Complex behavior can be simulated in real time such as friction along the shaft of the needle, puncture and cutting force resulting from interactions of the needle with the tissue. The method is illustrated both in an interactive simulation of a needle insertion/cutting and in a robotic needle insertion in liver tissue during the breathing motion.