Indentation stiffness tomography of fibrous inhomogeneities — An asymptotic model

AFM (Atomic Force Microscopy)-based stiffness tomography poses novel indentation problems for a heterogeneous elastic sample containing a single or multiple heterogeneities. A relatively stiff infinite elastic fiber buried in an elastic half-space represents a simple model for fibrous organelles inside a living cell. The leading-order asymptotic model for the frictionless unilateral indentation is constructed using the method of matched asymptotic expansions under the assumption that the diameter of the contact area is small compared to the depth of the fiber below the surface subjected to the indentation imaging. The fiber deformation is described in the framework of Euler’s theory of bending. The resulting system of integro-differential equations is solved by means of the Fourier transform. The approximate relation between the indenter displacement and the contact force is derived in explicit form. The fiber influence factor is introduced to evaluate the incremental indentation stiffness.