Finite element analysis of elastic-plastic solids under Vickers indentation: surface deformation

Document Type: Research Paper


Shiraz University


Finite element modeling has been used to study the development of surface deformation during indentation with a Vickers indenter. A wide range of materials with different elastic modulus and yield stresses are examined. Results show that in a pyramidal indentation process, for a perfectly plastic material, sinking-in during loading can change to pile-up in unloading. This phenomenon depends on the elastic modulus to yield stress ratio. Results also show that the amount of pile-up cannot be related solely to the strain-hardening exponent, as often assumed. Rather, after initially sinking-in at small depths of penetration, the pile-up for many materials evolves and increases gradually as the indenter is driven into the material. It is shown that the ratio of the plastic volume radius to the indentation depth is nearly constant during loading and it is a function of the yield stress and the Young modulus. Experimental verification in loading and unloading is carried out with the results of Alcala et al. (Acta Materialia, 2000, pp. 3451).


[1] O'Neil, H., Hardness measurements of metals and alloys. New Jersey: Chapman Hall, (1951).
[2] Tabor, D., The hardness of metals. Oxford: Clarendon Press, (1951).
[3] E. Söederlund and D.J. Rowcliffe, J Hard Mater., 5 (1994) 149.
[4] G. Pintaúde, M. G. di, V. Cuppari, C. G. Schön, A. Sinatora, and R. M. Souza, Zeitschrift für Metallkunde, 96 (2005) 1252.
[5] J. Alcala, A.E. Giannakopoulos and S. Suresh, J. Mater. Res 13 (1998) 1390.
[6] A.K. Bhattacharya and W.D. Nix, Int. J. Solids Structures, 27 (1991) 1047.
[7] T.A. Laursen and J.C. Simo, J. Mater. Res., 7 (1992) 618.
[8] V. Marx and H. Balke, Acta mater, 45 (1997) 3791.
[9] A. Bolshakov and G.M. Pharr, J. Mater. Res., 13 (1998) 1049.
[10] A.E. Giannakopoulos, P.L. Larsson and Vestergaard, R., Int. J. Solids Structures, 31 (1994) 2679.
[11] P.L. Larsson, A.E. Giannakopoulos, E. Söerlund, D.J. Rowcliffe, and R. Vestergaard, Int. J. Solids Structures, 33 (1996).
[12] A.E. Giannakopoulos and P.L. Larsson, Mech. Mater., 25 (1997) 1.
[13] J. Alcala, A.C. Barone, A.C. and M. Anglada, M., Acta mater., 48 (2000) 3451.
[14] J.M. Antunes, L.F., Menezes, and J.V. Fernandes, Key Eng. Mat., 230 (2002) 525.
[15] J.M. Antunes, L.F., Menezes and J.V. Fernandes, Int. J. of Solids & Struct., Vol. 43 (2006) 781.
[16] A. Millard, Castem 2000, Rapport D.E.M.T./92 300, C.E.A, (1992).
[17] S. Carlsson, S. Biwa and P.L. Larsson, Int. J. Mech. Sci., 42 (2000) 107.
[18] A.E. Giannakopoulos, P.L. Larsson, R. Vestegaard, Int. J. Solids Struct., 31 (1994) 2670.
[19] K.L. Johnson, J. of the Mech. and Phys. of Solids, 18 (1970) 115.
[20] K.L. Johnson, Contact mechanics, Cambridge University Press, Cambridge, UK (1985).
[21] R.K. Abu Al-Rub, Mech. of Mat., 39 (2007) 787.