Analytical model for stick–slip phenomenon in solid tubular expansion

Solid tubular expansion is a metal forming process in which the inner diameter of a tube is increased to a desired value by forcing a conical mandrel through it. Large friction takes place at the mandrel/tubular interface during this operation. Typically, the static friction coefficient between two surfaces in contact is larger than the kinetic friction coefficient. If an applied force is large enough to overcome the static friction, then the reduction of the friction force to the kinetic value causes a sudden jump in the velocity of movement. This sticking and slipping of one part against the other is known as stick–slip and results in fluctuation in the force required for expansion as well as unexpected changes in length and thickness of the expanded tubular. A mathematical model depicting the dynamics of a stick–slip phenomenon in tube expansion has been developed. Three different sets of equations (one each for stick, slip, and transition phases) are derived using equilibrium equations, incompressibility condition and Karnopp׳s friction model. A zero velocity interval is used to define stick, slip and transition phases. A MATLAB program has been written to obtain an analytical solution using the developed governing equations. Comparison between experimental and analytical results shows good agreement for various parameters such as expansion force, thickness reduction and length shortening. The proposed model gives reasonably good prediction of the stick–slip behavior observed during experimental study. The fluctuation in the displacement–time plot clearly shows sticking of the mandrel. Subsequent slipping results in more thickness reduction which can reduce the structural integrity of the tube during its service life. Sensitivity analysis shows that mandrel velocity, friction coefficient, mandrel geometry, and expansion ratio affect the thickness reduction and the force required for expansion. A careful optimum selection of these parameters is important for enhanced performance of the expandable tubular during its service life.