Hostname: page-component-8448b6f56d-c4f8m Total loading time: 0 Render date: 2024-04-24T03:16:54.084Z Has data issue: false hasContentIssue false
  • English
  • Français

The correlation between the internal material length scale and the microstructure in nanoindentation experiments and simulations using the conventional mechanism-based strain gradient plasticity theory

Published online by Cambridge University Press:  31 January 2011

B. Backes ,
Y.Y. Huang ,
M. Göken  and
K. Durst
B. Backes*
Affiliation:
General Material Properties, University of Erlangen-Nürnberg, 91058 Erlangen, Germany
Y.Y. Huang
Affiliation:
Civil Engineering, Northwestern University of Chicago, Evanston, Illinois 60208-3109
M. Göken
Affiliation:
General Material Properties, University of Erlangen-Nürnberg, 91058 Erlangen, Germany
K. Durst
Affiliation:
General Material Properties, University of Erlangen-Nürnberg, 91058 Erlangen, Germany
*
a) Address all correspondence to this author. e-mail: bjoern.backes@ww.uni-erlangen.de
Get access

Abstract

In the present work a new equation to determine the internal material length scale for strain gradient plasticity theories from two independent experiments (uniaxial and nanoindentation tests) is introduced. The applicability of the presented equation is verified for conventional grained as well as for ultrafine-grained copper and brass with different amounts of prestraining. A significant decrease of the experimentally determined internal material length scale is found for increasing dislocation densities due to prestraining. Conventional mechanism strain gradient plasticity theory is used for simulating the indentation response, using experimentally determined material input data as the yield stress, the work-hardening behavior and the internal material length scale. The work-hardening behavior and the yield stress were taken from the uniaxial experiments, whereas the internal material length scale was calculated using the yield stress from the uniaxial experiment, the macroscopic hardness H0 and the length scale parameter h* following from the nanoindentation experiment. A good agreement between the measured and calculated load–displacement curve and the hardness is found independent of the material and the microstructure.

Keywords

HardnessNanoindentationNanoscale
Type
Articles
Information
Journal of Materials Research , Volume 24 , Issue 3 , March 2009 , pp. 1197 - 1207
Copyright
Copyright © Materials Research Society 2009

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

1.Nix, W.D. and Mehl, M.R.: Mechanical properties of thin films. Metall. Trans. A 20, 2217 (1989).CrossRefGoogle Scholar
2.De Guzman, M.S., Neubauer, G., Flinn, P., and Nix, W.D.: The role of indentation depth on the measured hardness of materials, in Thin Films: Stresses and Mechanical Properties IV, edited by Townsend, P.H., Weihs, T.P., Sanchez, J.E. Jr, and Borgesen, P. (Mater. Res. Soc. Symp. Proc. 308, Pittsburgh, PA, 1993), p. 613.Google Scholar
3.Stelmashenko, N.A., Walls, M.G., Brown, L.M., and Milman, Y.V.: Microindentation on W and Mo oriented single crystals: An STM study. Acta Mater. 41, 2855 (1993).CrossRefGoogle Scholar
4.Ma, Q. and Clarke, D.R.: Size dependent hardness of silver single crystals. J. Mater. Res. 10, 853 (1995).CrossRefGoogle Scholar
5.McElhaney, K.W., Vlassak, J.J., and Nix, W.D.: Determination of indenter tip geometry and indentation contact area for depth-sensing indentation experiments., J. Mater. Res. 13, 1300 (1998).CrossRefGoogle Scholar
6.Suresh, S., Nieh, T.G., and Choi, B.W.: Nano-indentation of copper thin films on silicon substrates. Scr. Mater. 41, 951 (1999).CrossRefGoogle Scholar
7.Durst, K., Backes, B., and Göken, M.: Indentation size effect in metallic materials: Correcting for the size of the plastic zone. Scr. Mater. 52, 1093 (2005).CrossRefGoogle Scholar
8.Durst, K., Backes, B., Franke, O., and Göken, M.: Indentation size effect in metallic materials: Modeling strength from pop-in to macroscopic hardness using geometrically necessary dislocations. Acta Mater. 54, 2547 (2006).CrossRefGoogle Scholar
9.Fleck, N.A., Muller, G.M., Ashby, M.F., and Hutchinson, J.W.: Strain gradient plasticity: Theory and experiment. Acta Metall. Mater. 42, 475 (1994).CrossRefGoogle Scholar
10.Stölken, J.S. and Evans, A.G.: A microbend test method for measuring the plasticity length scale. Acta Mater. 46, 5109 (1998).CrossRefGoogle Scholar
11.Motz, C., Schöberl, T., and Pippan, R.: Mechanical properties of micro-sized copper bending beams machined by the focused ion beam technique. Acta Mater. 53, 4269 (2005).CrossRefGoogle Scholar
12.Nye, J.: Some geometrical relation in dislocated crystals. Acta Metall. Mater. 1, 153 (1953).CrossRefGoogle Scholar
13.Cottrell, A.H.: The Mechanical Properties of Materials (Wiley, New York, 1964), p. 277.Google Scholar
14.Ashby, M.F.: The deformation of plastically non-homogeneous alloys. Philos. Mag. 21, 399 (1970).CrossRefGoogle Scholar
15.Fleck, N.A. and Hutchinson, J.W.: A phenomenological theory for strain gradient effects in plasticity. J. Mech. Phys. Solids 41, 1825 (1993).CrossRefGoogle Scholar
16.Taylor, G.I.: The mechanism of plastic deformation of crystals. Part I. Theoretical. Proc. R. Soc. London Ser. A 145, 362 (1934).Google Scholar
17.Duan, D.M., Wu, N.Q., Slaughter, W.S., and Mao, S.X.: Length scale effect on mechanical behavior due to strain gradient plasticity. Mater. Sci. Eng., A 303, 241 (2001).CrossRefGoogle Scholar
18.Abu Al-Rub, R.K. and Voyiadjis, G.Z.: Analytical and experimental determination of the material intrinsic length scale of strain gradient plasticity theory from micro- and nano-indentation experiments. Int. J. Plast. 20, 1139 (2004).CrossRefGoogle Scholar
19.Abu Al-Rub, R. and Voyiadjis, G.: A physically based gradient plasticity. Int. J. Plast. 22, 654 (2006).CrossRefGoogle Scholar
20.Abu Al-Rub, R.: Prediction of micro and nanoindentation size effect from conical or pyramidal indentation. Mech. Mater. 39, 787 (2007).CrossRefGoogle Scholar
21.Mughrabi, H.: On the current understanding of strain gradient plasticity. Mater. Sci. Eng., A 387–389, 209 (2004).CrossRefGoogle Scholar
22.Sadrabadi, P., Durst, K., and Göken, M.: Study on the indentation size effect in CaF2: Dislocation structure and hardness. Acta Mater. (2008).Google Scholar
23.Gao, H., Huang, Y., Nix, W.D., and Hutchinson, J.W.: Mechanism-based strain gradient plasticity—I. Theory. J. Mech. Phys. Solids 47, 1239 (1999).CrossRefGoogle Scholar
24.Chen, S.H. and Wang, T.C.: A new hardening law for strain gradient plasticity. Acta Mater. 48, 3997 (2000).CrossRefGoogle Scholar
25.Chen, S.H. and Wang, T.C.: A new deformation theory with strain gradient effects. Int. J. Plast. 18, 971 (2002).CrossRefGoogle Scholar
26.Huang, Y., Gao, H., Nix, W.D., and Hutchinson, J.W.: Mechanism-based strain gradient plasticity. II. Analysis. J. Mech. Phys. Solids 48, 99 (2000).CrossRefGoogle Scholar
27.Huang, Y., Xue, Z., Gao, H., Nix, W.D., and Xia, Z.C.: A study of microindentation hardness tests by mechanism-based strain gradient plasticity. J. Mater. Res. 15, 1786 (2000).CrossRefGoogle Scholar
28.Qiu, X., Huang, Y., Nix, W.D., Hwang, K.C., and Gao, H.: Effect of intrinsic lattice resistance in strain gradient plasticity. Acta Mater. 49, 3949 (2001).CrossRefGoogle Scholar
29.Qiu, X., Huang, Y., Wei, Y., Gao, H., and Hwang, K.C.: The flow theory of mechanism-based strain gradient plasticity. Mech. Mater. 35, 245 (2003).CrossRefGoogle Scholar
30.Gurtin, M.E.: On the plasticity of single crystals: Free energy, microforces, plastic-strain gradient. J. Mech. Phys. Solids 48, 989 (2000).CrossRefGoogle Scholar
31.Gurtin, M.E.: A gradient theory of single-crystal viscoplasticity that accounts for geometrically necessary dislocations. J. Mech. Phys. Solids 50, 5 (2002).CrossRefGoogle Scholar
32.Hwang, K.C., Jiang, H., Huang, Y., Gao, H., and Hu, N.: A finite deformation theory of strain gradient plasticity. J. Mech. Phys. Solids 50, 81 (2002).CrossRefGoogle Scholar
33.Hwang, K.C., Jiang, H., Huang, Z., and Gao, H.: Finite deformation analysis of mechanism-based strain gradient plasticity: Torsion and crack tip field. Int. J. Plast. 19, 235 (2003).CrossRefGoogle Scholar
34.Wang, W., Huang, Y., Hsia, K.J., Hu, K.X., and Chandra, A.: A study of microbend test by strain gradient plasticity. Int. J. Plast. 19, 365 (2003).CrossRefGoogle Scholar
35.Acharya, A. and Bassani, J.L.: Lattice incompatibility and a gradient theory of crystal plasticity. J. Mech. Phys. Solids 48, 1565 (2000).CrossRefGoogle Scholar
36.Acharya, A. and Beaudoin, A.J.: Grain-size effect in viscoplastic polycrystals at moderate strains. J. Mech. Phys. Solids 48, 2213 (2000).CrossRefGoogle Scholar
37.Bassani, J.L.: Incompatibility and a simple gradient theory of plasticity. J. Mech. Phys. Solids 49, 1983 (2001).CrossRefGoogle Scholar
38.Beaudoin, A.J. and Acharya, A.: A model for rate-dependent flow of metal polycrystals based on the slip plane lattice incompatibility. Mater. Sci. Eng., A 309, 411 (2001).CrossRefGoogle Scholar
39.Evers, L.P., Parks, D.M., Brekelmans, W.A.M., and Geers, M.G.D.: Crystal plasticity model with enhanced hardening by geometrically necessary dislocation accumulation. J. Mech. Phys. Solids 50, 2403 (2002).CrossRefGoogle Scholar
40.Huang, Y., Qu, S., Hwang, K.C., Li, M., and Gao, H.: A conventional theory of mechanism-based strain gradient plasticity. Int. J. Plast. 20, 753 (2004).CrossRefGoogle Scholar
41.Lee, H., Ko, S., Han, J., Park, H., and Hwang, W.: Novel analysis for nanoindnentaion size effect using strain gradient plasticity. Scr. Mater. 53, 1135 (2005).CrossRefGoogle Scholar
42.Nix, W.D. and Gao, H.: Indentation size effect in crystalline materials: A law for strain gradient plasticity. J. Mech. Phys. Solids 46, 411 (1998).CrossRefGoogle Scholar
43.Huang, Y., Zhang, F., Hwang, K., Nix, W., Pharr, G., and Feng, G.: A model of size effects in nano-indentation. J. Mech. Phys. Solids 54, 1668 (2006).CrossRefGoogle Scholar
44.Durst, K., Franke, O., Bohner, A., and Göken, M.: Indentation size effect in Ni-Fe solid solutions. Acta Mater. 55, 6825 (2007).CrossRefGoogle Scholar
45.Pharr, G.M.: Conference on the Indentation Behavior of Materials, Hyderabad, India (2008).Google Scholar
46.Tho, K.K., Swaddiwudhipong, S., Hua, J., and Liu, Z.S.: Numerical simulation of indentation with size effect. Mater. Sci. Eng. 421, 268 (2004).CrossRefGoogle Scholar
47.Qu, S., Huang, Y., Pharr, G.M., and Hwang, K.C.: The indentation size effect in the spherical indentation of iridium: A study via the conventional theory of mechanism-based strain gradient plasticity. Int. J. Plast. 22, 1265 (2006).CrossRefGoogle Scholar
48.Arsenlis, B.A. and Parks, D.M.: Crystallographic aspects of geometrically-necessary and statistically-stored dislocation density. Acta Mater. 47, 1597 (1999).CrossRefGoogle Scholar
49.Bishop, J.F.W. and Hill, R.: A theory of plastic distortion of a polycrystalline aggregate under combined stresses. Philos. Mag. 42, 414 (1951).CrossRefGoogle Scholar
50.Bishop, J.F.W. and Hill, R.: A theoretical derivation of the plastic properties of a polycrystalline face-centered metal. Philos. Mag. 42, 1298 (1951).CrossRefGoogle Scholar
51.Kocks, U.F.: The relation between polycrystal deformation and single crystal deformation. Metall. Mater. Trans. 1, 1121 (1970).CrossRefGoogle Scholar
52.Höppel, H.W., Zhou, Z.M., Mughrabi, H., and Valiev, R.Z.: Microstructural study of the parameters governing coarsening and cyclic softening in fatigued ultrafine-grained copper. Philos. Mag. A 82, 1781 (2002).CrossRefGoogle Scholar
53.Mughrabi, H., Höppel, H.W., and Kautz, M.: Fatigue and microstructure of ultrafine-grained metals produced by severe plastic deformation. Scr. Mater. 51, 807 (2004).CrossRefGoogle Scholar
54.Backes, B., Durst, K., and Göken, M.: Determination of plastic properties of polycristalline metallic materials by nanoindentation: Experiments and finite element simulations. Philos. Mag. 86, 5541 (2006).CrossRefGoogle Scholar
55.Oliver, W.C. and Pharr, G.M.: An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments. J. Mater. Res. 7, 1564 (1992).CrossRefGoogle Scholar