Hostname: page-component-76fb5796d-5g6vh Total loading time: 0 Render date: 2024-04-26T14:36:54.370Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  23 September 2013

P. Sudharshan Phani ,
Kurt E. Johanns ,
Easo P. George  and
George M. Pharr
P. Sudharshan Phani
Affiliation:
Department of Materials Science and Engineering, The University of Tennessee, Knoxville, Tennessee 37996
Kurt E. Johanns
Affiliation:
Department of Materials Science and Engineering, The University of Tennessee, Knoxville, Tennessee 37996
Easo P. George
Affiliation:
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
George M. Pharr*
Affiliation:
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
*
a)Address all correspondence to this author. e-mail: pharr@utk.edu
Get access

Abstract

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.

Keywords

nanoindentationdislocationsstrength
Type
Articles
Information
Journal of Materials Research , Volume 28 , Issue 19 , 14 October 2013 , pp. 2728 - 2739
Copyright
Copyright © Materials Research Society 2013 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Bei, H., Shim, S., Pharr, G.M., and George, E.P.: Effects of pre-strain on the compressive stress–strain response of Mo-alloy single-crystal micropillars. Acta Mater. 56, 4762 (2008).CrossRefGoogle Scholar
Bei, H., Shim, S., George, E.P., Miller, M.K., Herbert, E.G., and Pharr, G.M.: Compressive strengths of molybdenum alloy micro-pillars prepared using a new technique. Scr. Mater. 57, 397 (2007).CrossRefGoogle Scholar
Uchic, M.D., Dimiduk, D.M., Florando, J.N., and Nix, W.D.: Sample dimensions influence strength and crystal plasticity. Science 305, 986 (2004).CrossRefGoogle ScholarPubMed
Uchic, M.D. and Dimiduk, D.A.: A methodology to investigate size scale effects in crystalline plasticity using uniaxial compression testing. Mater. Sci. Eng., A 400, 268 (2005).CrossRefGoogle Scholar
Greer, J.R., Oliver, W.C., and Nix, W.D.: Size dependence of mechanical properties of gold at the micron scale in the absence of strain gradients. Acta Mater. 53, 1821 (2005).CrossRefGoogle Scholar
Hemker, K.J. and Sharpe, W.N.: Microscale characterization of mechanical properties. Annu. Rev. Mater. Res. 37, 93 (2007).CrossRefGoogle Scholar
Kiener, D., Grosinger, W., Dehm, G., and Pippan, R.: A further step towards an understanding of size-dependent crystal plasticity: In situ tension experiments of miniaturized single-crystal copper samples. Acta Mater. 56, 580 (2008).CrossRefGoogle Scholar
Kiener, D., Motz, C., Schoberl, T., Jenko, M., and Dehm, G.: Determination of mechanical properties of copper at the micron scale. Adv. Eng. Mater. 8, 1119 (2006).CrossRefGoogle Scholar
Kraft, O., Gruber, P.A., Monig, R., and Weygand, D.: Plasticity in confined dimensions. Annu. Rev. Mater. Res. 40, 293 (2010).CrossRefGoogle Scholar
Schneider, A.S., Clark, B.G., Frick, C.P., Gruber, P.A., and Arzt, E.: Effect of orientation and loading rate on compression behavior of small-scale Mo pillars. Mater. Sci. Eng., A 508, 241 (2009).CrossRefGoogle Scholar
Volkert, C.A. and Lilleodden, E.T.: Size effects in the deformation of sub-micron Au columns. Philos. Mag. 86, 5567 (2006).CrossRefGoogle Scholar
Uchic, M.D., Dimiduk, D.M., Florando, J.N., and Nix, W.D.. Exploring specimen size effects in plastic deformation of Ni3(Al, Ta), in Defect Properties and Related Phenomena in Intermetallic Alloys, edited by E.P. George, H. Inui, M.J. Mills, and G. Eggler (Mater. Res. Soc. Symp. Proc. 753, Warrendale, PA, 2002).Google Scholar
Uchic, M.D., Shade, P.A., and Dimiduk, D.M.: Plasticity of micrometer-scale single crystals in compression. Annu. Rev. Mater. Res. 39, 361 (2009).CrossRefGoogle Scholar
Nix, W.D. and Gao, H.: Indentation size effects in crystalline materials: A law for strain gradient plasticity. J. Mech. Phys. Solids 46, 411 (1998).CrossRefGoogle Scholar
Swadener, J.G., George, E.P., and Pharr, G.M.: The correlation of the indentation size effect measured with indenters of various shapes. J. Mech. Phys. Solids 50, 681 (2002).CrossRefGoogle Scholar
Shim, S., Bei, H., George, E.P., and Pharr, G.M.: A different type of indentation size effect. Scr. Mater. 59, 1095 (2008).CrossRefGoogle Scholar
Phani, P.S., Johanns, K.E., Duscher, G., Gali, A., George, E.P., and Pharr, G.M.: Scanning transmission electron microscope observations of defects in as-grown and pre-strained Mo alloy fibers. Acta Mater. 59, 2172 (2011).CrossRefGoogle Scholar
Barnoush, A., Welsch, M.T., and Vehoff, H.: Correlation between dislocation density and pop-in phenomena in aluminum studied by nanoindentation and electron channeling contrast imaging. Scr. Mater. 63, 465 (2010).CrossRefGoogle Scholar
Lodes, M.A., Hartmaier, A., Goken, M., and Durst, K.: Influence of dislocation density on the pop-in behavior and indentation size effect in CaF2 single crystals: Experiments and molecular dynamics simulations. Acta Mater. 59, 4264 (2011).CrossRefGoogle Scholar
Barnoush, A.: Correlation between dislocation density and nanomechanical response during nanoindentation. Acta Mater. 60, 1268 (2012).CrossRefGoogle Scholar
Sekido, K., Ohmura, T., Hara, T., and Tsuzaki, K.: Effect of dislocation density on the initiation of plastic deformation on Fe-C steels. Mater. Trans. 53, 907 (2012).CrossRefGoogle Scholar
Lilleodden, E.T. and Nix, W.D.: Microstructural length-scale effects in the nanoindentation behavior of thin gold films. Acta Mater. 54, 1583 (2006).CrossRefGoogle Scholar
Morris, J.R., Bei, H., Pharr, G.M., and George, E.P.: Size effects and stochastic behavior of nanoindentation pop in. Phys. Rev. Lett. 106, 165502 (2011).CrossRefGoogle ScholarPubMed
Johanns, K.E., Sedlmayr, A., Sudharshan Phani, P., Mönig, R., Kraft, O., George, E.P., and Pharr, G.M.: In-situ tensile testing of single-crystal molybdenum-alloy fibers with various dislocation densities in a scanning electron microscope. J. Mater. Res. 27, 508 (2012).CrossRefGoogle Scholar
Li, T.L., Bei, H., Morris, J.R., George, E.P., and Gao, Y.F.: Scale effects in convoluted thermal/spatial statistics of plasticity initiation in small stressed volumes during nanoindentation. Mater. Sci. Technol. 28, 1055 (2012).CrossRefGoogle Scholar
Sudharshan Phani, P., Johanns, K.E., George, E.P., and Pharr, G.M.: A simple stochastic model for yielding in specimens with limited number of dislocations. Acta Mater. 61, 2489 (2013).CrossRefGoogle Scholar
Ngan, A.H.W., Zuo, L., and Wo, P.C.: Size dependence and stochastic nature of yield strength of micron-sized crystals: A case study on Ni3Al. Proc. R. Soc. Lond. A 462, 1661 (2006).Google Scholar
Parthasarathy, T.A., Rao, S.I., Dimiduk, D.M., Uchic, M.D., and Trinkle, D.R.: Contribution to size effect of yield strength from the stochastics of dislocation source lengths in finite samples. Scr. Mater. 56, 313 (2007).CrossRefGoogle Scholar
El-Awady, J.A., Wen, M., and Ghoniem, N.M.: The role of the weakest-link mechanism in controlling the plasticity of micropillars. J. Mech. Phys. Solids 57, 32 (2009).CrossRefGoogle Scholar
Ngan, A.H.W.: An explanation for the power-law scaling of size effect on strength in micro-specimens. Scr. Mater. 65, 978 (2011).CrossRefGoogle Scholar
Gu, R. and Ngan, A.H.W.: Dislocation arrangement in small crystal volumes determines power-law size dependence of yield strength. J. Mech. Phys. Solids 61, 1531 (2013).CrossRefGoogle Scholar
Fischer-Cripps, A.C.: Introduction to Contact Mechanics (Springer, New York, 2000).Google Scholar
Huber, M.T.: Zur Theorie der Berührung fester elastischer Körper. Annal. Phys. 319, 153 (1904).CrossRefGoogle Scholar
Dieter, G.E.: Mechanical Metallurgy (McGraw-Hill, New York, 1976).Google Scholar
Barabash, R.I.: Unpublished work.Google Scholar
Ogata, S., Li, J., Hirosaki, N., Shibutani, Y., and Yip, S.: Ideal shear strain of metals and ceramics. Phys. Rev. B 70, 104104 (2004).CrossRefGoogle Scholar