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Moving-contact problems of dynamic elasticity

Michele Brun (Università di Cagliari), Leonid Slepyan (School of Mechanical Engineering, Tel Aviv University.)

SES Medal Symposium in honor of D.J. Steigmann

Mon 2:40 - 4:00

MacMillan 115

We consider the problem of a rigid body moving in contact with an elastic medium [1]. The problem has been studied in the framework of linear elasticity and the general smooth frictionless solution has been obtained. We discuss the general conditions under which the rigid body meets zero driving force, drawing a parallelism with the d'Alembert paradox for incompressible and inviscid potential flow. The steady-state general solution is obtained as the limit for the related transient problem, which provides the necessary information regarding energy flux from infinity. Mathematically, the elastic solution is found solving a mixed problem for a single analytical function. The solution (displacement and stress) is based on the introduction of a new condition regarding energy fluxes at singular points in addition to the well-known Signorini conditions. For the half-plane contact problem the solution is given for any speed, in the subsonic (sub-Rayleigh and super-Rayleigh), intersonic and supersonic regimes. In the super-Rayleigh subsonic speed range, the leading point of the contact area appears as an energy absorbing singular point and a speed-dependent driving force is required for the motion. In the intersonic and supersonic speed regimes, the wave radiation creates resistance. Interestingly, no resistance for the movement is found for subsonic speeds, but also at the longitudinal wave speed. The related the steady-state problems of wedging of an elastic plane by a smooth rigid body and the movement of a finite rigid body along the interface of two elastic half-planes compressed together are also investigated in the sub-Rayleigh regimes. [1] Slepyan, L., Brun, M. 2012 “Driving forces in moving-contact problems of dynamics elasticity: indentation, wedging and free-sliding” J. Mech. Phys. Solids., 60, 11, 1883-1996.