Effects of surface tension on spherical indentation of viscoelastic materials during loading and unloading

Indentation serves as a fundamental technique to characterize the viscoelastic characteristics of soft biological materials including cells, tissues and hydrogels. However, the effect of surface tension on indentation responses of viscoelastic solids has received limited attention. In this study, we examine the spherical indentation of viscoelastic materials with surface tension during both loading and unloading scenario through finite element methods. For indentations under an intermediate velocity, results demonstrate that surface tension reduces the dimensionless normal displacement within and outside the contact zone, counteracting the increase of displacement induced by viscoelasticity, and the reduction becomes more pronounced with higher surface tension. Concurrently, surface tension redistributes normal stress from the contact center toward the periphery, smoothing stress gradients at the contact edge. Furthermore, surface tension increases the load required to achieve a given indentation depth and decreases the critical indentation depth at zero load, therefore delaying the detachment between the indenter and substrate. Under extreme (very low/high) loading velocities, elastic predictions with surface tension could accurately describe the load and contact radius. In contrast, at intermediate velocities, viscoelasticity exerts a more profound influence than surface tension, which leads to accelerated detachment and further reductions of both load and contact radius during late unloading stage. This study advances the understanding of surface tension effects on mechanical responses of indentation on biological materials.