Endless elastic beam travelling on a moving rough surface with zones of stick and sliding

We study the planar deformation of a beam that travels across a given control domain supported by a moving rough plane, which is a prototype for various technological processes. A sufficiently small misalignment between the guideways at the ends of the domain results in a stationary regime of motion, which features a zone of sticking contact near the entry to the control domain, followed by infinitely many segments of transverse sliding with alternating directions. Self-similarity of this solution of an essentially nonlinear boundary value problem is the primary novel result of the present contribution. Closed-form analytic results are validated against a finite element simulation of the transient evolution process, which demonstrates stability of the obtained solution and provides insights regarding the characteristic time scales of establishing of subsequent zones of sliding.