To determine the load-carrying capacity of the drillpipe slip, it is necessary to calculate the drag force on the surface between the pipe and the slip. To improve such capacity, the grip surface is provided with teeth that can indent into the pipe body. As a result, the friction force on the contact surface is supplemented with the drag force generated by plastic shear strain of the pipe body. The paper presents an analytic dependence of the indentation force of an ideal (untruncated) and non-ideal (truncated) wedge punch that models tooth operation on the indentation depth and friction factor on the punch lateral face both for shallow (with prevailing elastic strain of the gripped body) and deep indentation (with prevailing plastic strain). Multiple computational experiments were performed to identify parameters of the proposed formula. Such computations were complemented with determination of the punch drag force dependence on the indentation depth during punch shearing. The developed model was verified by experimental studies of punch indentation and slip as well as indentation and drag force measurements. The obtained results proved the high level of accuracy of the analytical indentation force model. The outcome of drag force determination experiments was further used to calculate the slip loadcarrying capacity.