On the depth of cylindrical indentation of an elastic half-space for two types of displacement constraints

Abstract

For cylindrical indentation of elastic half-space the relationship between the depth of indentation delta and the applied force F is nonlinear, in contrast to the linear relationship between the height of the contact zone delta_0 and the force F. While the latter is independent of the boundary conditions used to specify the rigid-body translation, the former depends on a selected datum for vertical displacement. The depth of the indentation is determined for any permissible value of the length b, which specifies the points of the free surface where the vertical displacement is required to be zero, w(b)=0. From the condition that the work of the indentation force is equal to the work of the contact pressure, it follows that the indentation is geometrically and physically possible under imposed boundary conditions w(b)=0 provided that b>=b_min. The numerical value of b_min is found to be about 10 times greater than the semi-width of the contact zone a, based on the numerical precision in fulfilling the work condition W_F=W_p. If a datum is taken to be at a point at some distance h below the load, there is an alternative closed-form expression for delta in terms of F, which involves the Poisson ratio nu. For nu=1/3, it is found that h_min is about 21a. A simple expression relating the permissible values of h and b is derived, which is linear for large values of h and b.