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The mathematical model of plastic deformation in an initially anisotropic medium assumes that principal directions of strain in elasticity remain principal directions of strain in plasticity and fracture as well. The modeling considers penetration of a bit represented by a cylindrical tool with a wedge-shaped tip. The maximal penetration depth of the bit is calculated at the preset initial rate of penetration. The influence of the anisotropy of the medium and the bit parameters on the bit penetration depth is studied.