Numerical simulation of a rigid body interacting with a viscoelastic soft material in a viscous fluid

Atomic force microscopy (AFM) is widely recognized as a state-of-the-art tool to investigate the mechanical properties of soft samples, especially in biological applications. Such measurements are often made in a fluid environment to mimic the natural biological surrounding. When the magnitude of the force is small, typically during the measurement of soft samples, the viscous force can be of a similar order as the reaction force from the sample, leading to large errors due to its neglect. Here we numerically investigate the force acting on a spherical AFM tip indenting on a viscoelastic substrate submerged in a fluid environment. We use a front tracking finite difference method to solve the flow around a spherical probe moving with a constant velocity towards a viscoelastic solid substrate. The solid sphere is modeled using a moving least square (MLS) immersed boundary method (IBM). The viscoelastic substrate is modeled with two non-linear constitutive equations, first combining a neo-Hookean spring with a dashpot in parallel (a non-linear Kelvin-Voigt type model) and second a neo-Hookean spring with an upper convective Maxwell element in parallel. We investigate the effects of fluid viscosity and substrate viscoelasticity on its deformation and the force on the probe.