A stochastic model for the size dependence of spherical indentation pop-in
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Easo P. George and George M. Pharra) Department of Materials Science and Engineering, The University of Tennessee, Knoxville, Tennessee 37996; and Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831 (Received 8 May 2013; accepted 15 July 2013)
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.
Address all correspondence to this author. e-mail: [email protected] This author was an editor of this journal during the review and decision stage. For the JMR policy on review and publication of manuscripts authored by editors, please refer to http://www.mrs. org/jmr-editor-manuscripts/. DOI: 10.1557/jmr.2013.254
flow, this phenomenon is the indentation equivalent of yielding in pillar testing. Hence, some of the same basic ideas used in understanding the size effect in uniaxial testing can also be applied to indentation testing. In pillar testing of single-phase materials, there are two important length scales that influence the plastic deformation behavior: (i) the size of the pillar and (ii) the average spacing of the dislocations it contains. It is usually when these two length scales are similar that the unusual size effects on strength become important.1 In addition to a size effect, the strength of small pillars tends to be very scattered and stochastic17 presumably because the yield and flow behavior are controlled by a small number of dislocations, the character, orientation, and mobility of which can vary significantly from one pillar to the next. An analogous size effect and stochastic behavior is observed in nanoindentation pop-in. As shown in Fig. 1, here again there are two important length scales: (i) the average spacing of dislocations, kavg 5 q1/2, where q is the dislocation density and (ii) the size of the highly stressed zone under the contact, which itself would scale with the radius of the indenter, R, and depend on the indentation load, F, and the elastic modulus, E. Although the primary geometric effect we will focus on here is the indenter radius effect, it should be no
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