A stochastic model for the size dependence of spherical indentation pop-in

A simple stochastic model is developed to determine the pop-in load and maximum shear stress at pop-in in nanoindentation experiments conducted with spherical indenters that accounts for recent experimental observations of a dependence of these parameters on the indenter radius. The model incorporates two separate mechanisms: pop-in due to nucleation of dislocations in dislocation-free regions and pop-in by activation of preexisting dislocations. Two different types of randomness are used to model the stochastic behavior, which include randomness in the spatial location of the dislocations beneath the indenter and randomness in the orientation of the dislocations, i.e., randomness in the stress needed to activate them. In addition to correctly predicting the experimentally observed average maximum shear stress at pop-in, the model also correctly describes the scatter in pop-in loads and how it varies with indenter radius. Monte Carlo simulations are used to validate the model and visualize the scatter expected for a limited number of tests.