On unilateral indentation problems encountered in AFM testing of living cells

Abstract A heuristic approach is employed to study unilateral indentation problems arising in the mechanical testing of living cells by means of atomic force microscopy (AFM). Interpreting the augmented Lagrangian formulation of a quasi-variational inequality as representing the effect of a Winkler-type coating, we propose a variational formulation—presently lacking a formal proof—for the indentation of an elastic substrate covered with a nonlinearly deforming, brush-like layer modeling the cell’s pericellular coat. In the absence of a rigorous justification, we introduce a three-parameter quasi-variational inequality and derive a dual form of Mossakovskii’s theorem. Building on the Itou-Kovtunenko-Rajagopal general solution to the unilateral indentation problem for a viscoelastic substrate with a non-increasing contact area, we obtain the displacement-force relation for monomial (axisymmetric) and self-similar (non-axisymmetric) indenters.