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<p>A&#xa0;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&#xa0;quasi-variational inequality as representing the effect of a&#xa0;Winkler-type coating, we propose a&#xa0;variational formulation—presently lacking a&#xa0;formal proof—for the indentation of an elastic substrate covered with a&#xa0;nonlinearly deforming, brush-like layer modeling the cell’s pericellular coat. In the absence of a&#xa0;rigorous justification, we introduce a&#xa0;three-parameter quasi-variational inequality and derive a&#xa0;dual form of Mossakovskii’s theorem. Building on the Itou-Kovtunenko-Rajagopal general solution to the unilateral indentation problem for a&#xa0;viscoelastic substrate with a&#xa0;non-increasing contact area, we obtain the displacement-force relation for monomial (axisymmetric) and self-similar (non-axisymmetric) indenters.</p>

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On unilateral indentation problems encountered in AFM testing of living cells

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  • 发表日期：

    2026年4月22日

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  • Ivan Argatov

摘要

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.