Interpretations of Indentation Size Effects

[+] Author and Article Information
W. W. Gerberich, N. I. Tymiak, J. C. Grunlan

Department of Chemical Engineering and Materials Science, University of Minnesota, 421 Washington Avenue, S.E., Minneapolis, MN 55455

M. F. Horstemeyer

Center for Materials and Engineering Sciences, Sandia National Laboratories, MS 9404, Livermore, CA 94551-0969

M. I. Baskes

Los Alamos National Laboratory, MSG 755, Los Alamos, NM 87545

J. Appl. Mech 69(4), 433-442 (Jun 20, 2002) (10 pages) doi:10.1115/1.1469004 History: Revised January 08, 2001; Received March 15, 2001; Online June 20, 2002
Copyright © 2002 by ASME
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Fleck,  N. A., and Hutchinson,  J. W., 1993, “A Phenomenological Theory for Strain Gradient Effects in Plasticity,” J. Mech. Phys. Solids, 41, pp. 1825–1857.
Ma,  Q., and Clarke,  D. R., 1995, “Size Dependent Hardness of Silver Single Crystals,” J. Mater. Res., 10(4), pp. 853–863.
Nix,  W. D., and Gao,  H., 1998, “Indentation Size Effects in Crystalline Materials: A Law for Strain Gradient Plasticity,” J. Mech. Phys. Solids, 46(3), pp. 411–425.
Aifantis,  E. C., 1984, “On the Microstructural Origin of Certain Inelastic Modes,” ASME J. Eng. Mater. Technol., 106, pp. 326–330.
Zbib,  H., and Aifantis,  E. C., 1989, “On the Localization and Post Localization of Plastic Deformation—Part I: On the Initiation of Shear Bands,” Res. Mech., pp. 261–277.
Zbib,  H., and Aifantis,  E. C., 1989, “On the Localization and Post Localization of Plastic Deformation—Part II: On the Evolution and Thickness of Shear Bands,” Res. Mech., pp. 279–292.
Zbib,  H., and Aifantis,  E. C., 1989, “On the Localization and Post Localization of Plastic Deformation—Part III: On the Structure and Velocity of Postevin–Le Chatelier Bands,” Res. Mech., pp. 293–305.
Shu,  J. Y., and Fleck,  N. A., 1998, “The Prediction of a Size Effect in Microindentation,” Int. J. Solids Struct., 35(13), pp. 1363–1383.
Gao,  H., Huang,  Y., Nix,  W. D., and Hutchinson,  J. W., 1999, “Mechanism-Based Strain Gradient Plasticity—I. Theory,” J. Mech. Phys. Solids, 47, pp. 1239–1263.
Hutchinson,  J. W., 2000, “Plasticity at the Micron Scale,” Int. J. Solids Struct., 37, pp. 225–238.
Ashby,  M. F., 1970, “The Deformation of Plastically Non-Homogeneous Alloys,” Philos. Mag., 21, pp. 399–424.
Poole,  W. J., Ashby,  M. F., and Fleck,  N. A., 1996, “Micro-Hardness Tests on Annealed and Work-Hardened Copper Polycrystals,” Scr. Metall., 34(4), pp. 559–564.
Wahl, K., and Asif, S. A., 2000, Naval Research Laboratories, personal communication.
Johnson,  K. L., Kendall,  K., and Roberts,  A. D., 1971, “Surface Energy and the Contact of Elastic Solids,” Proc. R. Soc. London, Ser. A, A321, pp. 301–313.
Derjaguin,  B. V., Muller,  V. M., and Toporov,  Yu. P., 1975, “Effect of Contact Deformations on the Adhesion of Particles,” J. Colloid Interface Sci., 53, pp. 314–326.
Maugis, D., 1999, “Contact, Adhesion and Rupture of Elastic Solids,” Series in Solid State Sciences, Springer, New York, pp. 62–66, 283–295.
Tymiak,  N. I., Kramer,  D. E., Bahr,  D. F., and Gerberich,  W. W., 2001, “Plastic Strain and Strain Gradients at Very Small Penetration Depths,” Acta Mater., 49, pp. 1021–1034.
Baskes, M., and Horstemeyer, M., 1999, Sandia National Labs, private communication.
Horstemeyer,  M. F., and Baskes,  M. I., 1999, “Atomistic Finite Deformation Simulations: A Discussion on Length Scale Effects in Relation to Mechanical Stresses,” ASME J. Eng. Mater. Technol., 121, pp. 114–119.
Gane,  M., and Cox,  J. M., 1970, “The Micro-Hardness of Metals at Very Low Loads,” Philos. Mag., 22(179), pp. 881–891.
Oliver,  W. C., and Pharr,  G. M., 1992, “An Improved Technique for Determining Hardness and Elastic Modulus Using Load and Displacement Sensing Experiments,” J. Mater. Res., 7(6), pp. 1564–1583.
Stelmashenko,  N. A., Walls,  M. G., Brown,  L. M., and Milman,  Yu. V., 1993, “Microindentations on W and Mo Oriented Single Crystals: An STM Study,” Acta Metall. Mater., 41(1), pp. 2855–2865.
Bahr,  D. F., Kramer,  D. E., and Gerberich,  W. W., 1998, “Non-Linear Deformation Mechanisms During Nanoindentation,” Acta Mater., 46(10), pp. 3605–3617.
Gerberich,  W. W., Kramer,  D. E., Tymiak,  N. I., Volinsky,  A. A., Bahr,  D. F., and Kriese,  M. D., 1999, “Nanoindentation-Induced Defect-Interface Interactions: Phenomena, Methods and Limitations,” Acta Mater., 47(15), pp. 4115–4123.
Bobji,  M. S., Biswas,  S. K., and Pethica,  J. B., 1997, “Effect of Roughness on the Measurement of Nanohardness—A Computer Simulation Study,” Appl. Phys. Lett., 71(8), pp. 1059–1061.
Gerberich,  W. W., Yu,  W., Kramer,  D., Strojny,  A., Bahr,  D., Lilleodden,  E., and Nelson,  J., 1998, “Elastic Loading and Elastoplastic Unloading from Nanometer Level Indentations for Modulus Determinations,” J. Mater. Res., 13, pp. 421–439.
Begley,  M. R., and Hutchinson,  J. W., 1998, “The Mechanics of Size-Dependent Indentation,” J. Mech. Phys. Solids, 46, pp. 2049–2068.
Belak, J., and Stowers, I. F., 1992, “The Indentation and Scraping of a Metal Surface: A Molecular Dynamics Study,” Fundamentals of Friction, I. L. Singer and H. M. Pollock, eds., Kluwer Academic, Dordrecht, pp. 511–520.
Johnson, K. L., 1985, Contact Mechanics, Cambridge Univ., Cambridge, UK, Press, pp. 153–184.
Kramer,  D., Huang,  H., Kriese,  M., Robach,  J., Nelson,  J., Wright,  A., Bahr,  D., and Gerberich,  W. W., 1999, “Yield Strength Predictions from the Plastic Zone Around Nanocontacts,” Acta Mater., 47(1), pp. 333–343.
Binnig,  G., Quate,  C. F., and Gerber,  Ch., 1987, “Atomic Force Microscope,” Phys. Rev. Lett., 56, pp. 930–933.
Kheshgi,  H. S., and Scriven,  L. E., 1991, “Dewetting, Nucleation and Growth of Dry Regions,” Chem. Eng. Sci., 46, pp. 519–526.
Josell,  D., and Spaepen,  F., 1993, “Determination of the Interfacial Tension by Zero Creep Experiments on Multilayers,” Acta Metall., 41, pp. 3015–3027.
Vermaak,  J. S., and Kuhlman-Wilsdorf,  D., 1968, “Measurement of the Average Surface Stress of Gold as a Function of Temperature in the Temperature Range 50–985°,” J. Phys. Chem., 72, pp. 4150–4154.
Mays,  C. W., Vermaak,  J. S., and Kuhlman-Wilsdorf,  D., 1968, “Surface Stress and Surface Tension. II. Determination of the Surface Stress of Gold,” Surf. Sci., 12, pp. 134–140.
Wasserman,  H. J., and Vermaak,  J. S., 1970, “Determination of a Lattice Contraction in Very Small Silver Particles,” Surf. Sci., 22, pp. 164–172.
Friesen, C., Dimitrov, N., Cammarata, R. C., and Sieradzki, K., 2000, “Surface Stress and the Electrocapilarity of Solid Electrodes,” Surf. Sci., submitted for publication.
Gerberich,  W. W., Venkataraman,  S. K., Huang,  H., Harvey,  S. E., and Kohlstedt,  D. L., 1995, “The Injection of Plasticity by Millinewton Contacts,” Acta Metall. Mater., 43(4), pp. 1569–1576.
Michalske,  T. A., and Houston,  J. E., 1998, “Dislocation Nucleation at Nano-Scale Mechanical Contacts,” Acta Mater., 46(2), pp. 391–396.
Gerberich,  W. W., Nelson,  J. C., Lilleodden,  E. T., Anderson,  P., and Wyrobek,  J. T., 1996, “Indentation Induced Dislocation Nucleation: The Initial Yield Point,” Acta Mater., 44(9), pp. 3585–3598.
Fleck,  N. A., Muller,  G. M., Ashby,  M. F., and Hutchinson,  J. W., 1994, Acta Metall. Mater., 42(2), pp. 475–487.
Gouldstone,  A., Koh,  H.-J., Zeng,  K.-Y., Giannakopolous,  A. E., and Suresh,  S., 2000, “Discrete and Continuous Deformation During Nanoindentation of Thin Films,” Acta Mater., 48, pp. 2277–2295.
Yasuda, K., Shinohara, K., Kinoshita, C. and Arai, M., 1994, “An Interpretation of the Indentation Size/Load Effect on Diamond Pyramid Hardness,” Strength of Materials, Oikawa et al., eds., The Japan Institute of Metal, pp. 865–868.
Corcoran,  S. G., Colton,  R. J., Lilleodden,  E. T., and Gerberich,  W. W., 1997, “Anomalous Plastic Deformation of Surfaces: Nanoindentation of Gold Single Crystals,” Phys. Rev. B, 55(24), pp. 16057–16060.
Cheng, L., 1996, “Numerical Modeling of Indentation and Scratch Problems,” Ph.D. Thesis, University of Minnesota.
Couchman,  P. R., Jesser,  W. A., Kuhlmann-Wilsdorf,  D., and Hirth,  J. P., 1972, Surf. Sci., 33, pp. 429–436.
Kramer,  D. E., Yoder,  K. B., and Gerberich,  W. W., 2001, “Surface Constrained Plasticity: Oxide Rupture and the Yield Point Process,” Philos. Mag. A, 81(8), pp. 2033–2058.
Cheng,  Y.-T., and Cheng,  C.-M., 1998, “Relationships Between Hardness, Elastic Modulus, and the Work of Indentation,” Appl. Phys. Lett., 73(5), pp. 614–616.
Hirth, J., and Loethe, J., 1982, Theory of Dislocations, 2nd Ed., John Wiley and Sons, pp. 837–839.


Grahic Jump Location
Hardness as a function of contact dimension in 〈100〉 tungsten crystals showing the ISE
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Average strain gradients, χ, versus depth of penetration, δ, into 〈100〉 W (open symbols) and 〈100〉 Al (closed symbols) crystals. Four different diamond tip radii used in each case.
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Schematic of spherical and sharp wedge contacts showing difference in strain gradient dependence on contact shape
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Mises strain as a function of distance, r, from indenter tip for a three-dimensional finite difference numerical analysis
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Ratio of surface to volume works as a function of indentation depth into 〈100〉 Al and 〈100〉 Fe-3wt%Si single crystals
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Surface-to-volume ratio, defined by projected contact area to plastic volume, as a function of indentation depth for (a) 〈100〉 W and 〈100〉 Fe-3wt%Si; (b) 〈100〉 Au and 〈100〉 Al single crystals. Solid and dashed curves represent the mean S/V values for each material. Note the different scales for the two materials in (b).
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Fit of the proposed model (Eq. (19)) for four 〈100〉 oriented single crystals. Single tips of 205 nm and 70 nm radii used in (a) and (c), respectively; multiple spherical tips with radii noted used in (b) and (d)
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Master plot of Eq. (19) for all materials
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Incorporation of the Baskes and Horstemeyer definition of volume to surface ratio, (V/S)B, for comparison of atomistic simulations to the present data: solid line is a power-law fit with −0.38 slope similar to Eq. (21b)



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