An analytical solution of the rebound indentation problem for an isotropic linear viscoelastic layer loaded with a spherical punch

Abstract

A unilateral frictionless axisymmetric contact problem for an isotropic viscoelastic layer attached to a rigid substrate and loaded with a spherical indenter is considered. It is assumed that the indentation protocol is composed of two stages. In the indentation phase, the layer is subjected to displacement loading, while at the end of the first stage, the load is immediately removed and the second stage, called the recovery phase, lasts for a theoretically indefinite time. Under the assumption of time-independent Poisson's ratio, we derive closed-form analytical expressions for the contact force (in the indentation phase) and for the indentation displacement (in the the recovery phase). The obtained closed-form analytical solution is valid for the indentation phase with an arbitrary monotonic loading displacement and can be used for evaluation of the rebound indentation test for soft biological tissues and originally suggested for assessment of articular cartilage viability.

Rebound indentation problem for a viscoelastic half‐space and axisymmetric indenter — Solution by the method of dimensionality reduction

“…It is interesting to observe that the rebound indentation displacement does not depend on the indenter shape, which, together with the fact that in the case of viscoelastic layer it does not depend on the layer thickness (this feature of the rebound displacement, which is measured after the removal of loading, was rigorously established in [12]), means that the rebound indentation represents a robust indicator of the viscoelastic response of a tested time-dependent material. In other words, this property of being insensitive to the indenter shape and the sample size factors is found to be crucial for the reliability of indentation-based testing for biological materials [17].…”

“…Recently, the rebound indentation problem was studied in the cases of cylindrical [11] and spherical [12] indenters. Here we generalize the obtained solution [12] for the case of arbitrary axisymmetric indenter, which produces a circular contact area during the whole process of indentation. For the sake of simplicity, we assume that Poisson's ratio, ν, of the half-space material is time independent.…”

Material property analytical relations for the case of an AFM probe tapping a viscoelastic surface containing multiple characteristic times

“…In other words, we have assumed that each time the cantilever taps on the sample, it finds it at a near rest position with negligible initial conditions (the displacement and force derivatives are assumed to be zero). The above makes the analytics tractable, although it is not formally true, since during sample recovery, the sample follows a trajectory related to the creep compliance function [ 36 , 46 – 47 ], and it is likely that before full recovery takes place the probe taps on the sample again (see the portion of the sample trajectory after the tip leaves the sample in Fig. 2 ).…”

Friction between a rigid body and a model elastomer having a linear viscous rheology

“…Adhesion is not in the scope of this study. The shear stresses σ are related to the deformation ratesε via 1 In experimental studies as the cited one the viscoelastic properties are often related to temperature variations instead of frequencies due to the experimental setup. But both approaches are closely connected via the master curve procedure [8].…”

“…In particular, the model is not suitable for thin layers of elastomers, which have to dealt with in a different way, cf. [1].…”