On the depth of cylindrical indentation of an elastic half-space for different types of displacement boundary conditions

Nonlinear relationships between the penetration depth δ and applied force F for cylindrical indentation of a half-space are derived corresponding to two types of displacement boundary conditions: (i) zero vertical displacement imposed at points x=±b of the free-surface of a half-space, and (ii) zero vertical displacement imposed at a point z=h below the indenter. The requirement that the work done by the indentation force must be equal to the work done by the contact pressure defines the minimum values of b and h for which the indentation is geometrically and physically possible. The minimum value of b is found to be about 10 times greater than the semi-width of the contact zone a=(4FR/πE∗)1/2, with the corresponding minimum value of the indentation depth δmin≈3.5δ0, where δ0=2F/πE∗ is the height of the contact zone under applied force F, and E∗=E/(1−ν2) is the plane strain elastic modulus. The minimum value of h is equal to about 21a in the case ν=1/3, and about 27a in the case ν=1/2. The relationship between h and b, corresponding to the same indentation depth δ≥δmin, is derived and shown to be nearly linear for large values of h/a and b/a. The obtained results are used to estimate the indentation depth of finite-size elastic bodies under different boundary constraints. Comparison with FEM prediction is made.