Load-displacement behavior during sharp indentation of viscous-elastic-plastic materials

A constitutive equation is developed for geometrically-similar sharp indentation of a material capable of elastic, viscous, and plastic deformation. The equation is based on a series of elements consisting of a quadratic (reversible) spring, a quadratic (time-dependent, reversible) dashpot, and a quadratic (time-independent, irreversible) slider-essentially modifying a model for an elastic-perfectly plastic material by incorporating a creeping component. Load-displacement solutions to the constitutive equation are obtained for load-controlled indentation during constant loading-rate testing. A characteristic of the responses is the appearance of a forward-displacing "nose" during unloading of load-controlled systems (e.g., magnetic-coil-driven "nanoindentation" systems). Even in the absence of this nose, and the associated initial negative unloading tangent, load-displacement traces (and hence inferred modulus and hardness values) are significantly perturbed on the addition of the viscous component. The viscous-elastic-plastic (VEP) model shows promise for obtaining material properties (elastic modulus, hardness, time-dependence) of time-dependent materials during indentation experiments.