Unified solution for poroelastic oscillation indentation on gels for spherical, conical and cylindrical indenters

An oscillation indentation method is developed for characterizing the local poroelastic properties of soft and hydrated materials such as hydrogels and biological tissues. In the dynamic oscillation indentation measurement, an indenter is pressed into the material to a certain depth and held for a period of time. After a plateau of force is reached, an oscillation of small depth is superimposed sweeping through a range of frequencies. The shift between the force and displacement spectra is denoted as the phase lag that characterizes the energy dissipative behavior of the soft hydrated materials due to solvent migration. A unified solution is obtained for the three widely used shapes of indenters for soft materials: cylindrical punch, spherical indenter and conical indenter. The solutions are summarized in remarkably simple forms allowing for easy extraction of material parameters including shear modulus, Poisson's ratio and diffusivity from the oscillation indentation measurements. The oscillation indentation measurement was demonstrated on a polyacrylamide (PAAm) gel using an atomic force microscope. It is shown that the time-dependent behavior of the PAAm gel at the micron scale is dominated by poroelasticity and the properties can be accurately extracted from the explicit expressions derived in this work. This method has great potential to be applied on heterogeneous biological tissues where local properties are of interest.