Abstract
A modified plastic damage model that accounts for tensile damage and compressive plasticity as well as interactions among them is adopted to simulate the indentation-induced cracking of silicon under Berkovich, cube corner, and Vickers indenters. Simulations with this model capture not only the well-known cracking geometries in indented ceramics, such as radial, median, lateral, and half penny (Vickers indenter) cracks, but also the recent experimentally discovered quarter penny cracks under Berkovich and cube corner pyramidal indenters. The quarter penny cracks are found to be formed by the coalescence of radial and median cracks for the first time in the simulation. Loads at which radial and half penny cracks are initiated in silicon are generally close to the experimental values reported in the literature, and the crack lengths on the sample surface agree well with both the current experimental measurements and analytical results by fracture mechanics.
Similar content being viewed by others
References
K.E. Petersen: Silicon as a mechanical material. Proc. IEEE 70 420 (1982)
R.F. Cook: Strength and sharp contact fracture of silicon. J. Mater. Sci. 41 841 (2006)
S. Johansson, J.A. Schweitz: Contact damage in single-crystalline silicon investigated by cross-sectional transmission electron microscopy. J. Am. Ceram. Soc. 71 617 (1988)
B.R. Lawn, D.B. Marshall, P. Chantikul: Mechanics of strength-degrading contact flaws in silicon. J. Mater. Sci. 16 1769 (1981)
M.G. Walls, M.M. Chaudhri, T.B. Tang: STM profilometry of low-load Vickers indentations in a silicon crystal. J. Phys. D: Appl. Phys. 25 500 (1992)
J.E. Bradby, J.S. Williams, J. Wong-Leung: Mechanical deformation in silicon by micro-indentation. J. Mater. Res. 16 1500 (2001)
H. Saka, A. Shimantani, M. Suganuma, M. Suprijadi: Transmission electron microscopy of amorphization and phase transformation beneath indents in Si. Philos. Mag. A 82 1971 (2002)
B.R. Lawn: Hertzian fracture in single crystals with the diamond. J. Appl. Phys. 39 4828 (1968)
A.B. Mann, D. van Heerden, J.B. Pethica, T.P. Weihs: The transformation of Si under point contacts. J. Mater. Res. 15 1754 (2000)
L. Zhang, I. Zarudi: Towards a deeper understanding of plastic deformation in mono-crystalline silicon. Int. J. Mech. Sci. 43 1985 (2001)
A.M. Minor, E.T. Lilleodden, M. Jin, E.A. Stach, D.C. Chrzan, J.W. Morris: Room temperature dislocation plasticity in silicon. Philos. Mag. 85 323 (2005)
B.R. Lawn, A.G. Evans: A model for crack initiation in elastic/plastic indentation fields. J. Mater. Sci. 12 2195 (1977)
B.R. Lawn, A.G. Evans, D.B. Marshall: Elastic/plastic indentation damage in ceramics: The median/radial crack system. J. Am. Ceram. Soc. 63 574 (1980)
S.S. Chiang, D.B. Marshall, A.G. Evans: The response of solids to elastic/plastic indentation. I. Stress and residual stresses. J. Appl. Phys. 53 298 (1982)
S.S. Chiang, D.B. Marshall, A.G. Evans: The response of solids to elastic/plastic indentation. II. Stress and residual stresses. J. Appl. Phys. 53 312 (1982)
E.H. Yoff: Elastic stress fields caused by indenting brittle materials. Philos. Mag. A 46 617 (1982)
K. Zeng, A.E. Giannakopoulos, D.J. Rowcliffe: Vickers indentations in glass. II. Comparison of finite element analysis and experiments. Acta Metall. Mater. 41 1945 (1995)
W. Zhang, G. Subhash: An elastic-plastic-cracking model for finite element analysis of indentation cracking in brittle materials. Int. J. Solids Struct. 38 5893 (2001)
A. Muchtar, L.C. Lim, K.H. Lee: Finite element analysis of Vickers indentation cracking processes in brittle solids using elements exhibiting cohesive post-failure behavior. J. Mater. Sci. 38 235 (2003)
J. Yan, A.M. Karlsson, X. Chen: On internal cone cracks induced by conical indentation in brittle materials. Eng. Fract. Mech. 74 2535 (2007)
A. Yonezu, B. Xu, X. Chen: Indentation induced lateral crack in ceramics with surface hardening. Mater. Sci. Eng., A 507 226 (2009)
A.E. Giannakopoulos, P.L. Larsson, R. Vestergaard: Analysis of Vickers indentation. Int. J. Solids Struct. 31 2679 (1994)
P.L. Larsson, A.E. Giannakopoulos: Analysis of Berkovich indentation. Int. J. Solids Struct. 33 221 (1996)
I. Zarudi, L.C. Zhang, W.C.D. Cheong, T.X. Yu: The difference of phase distributions in silicon after indentation with Berkovich and spherical indenters. Acta Mater. 53 479 (2005)
J.E. Bradby, J.S. Williams, J. Wong-Leung, M.V. Swain: Transmission electron microscopy observation of deformation microstructure under spherical indentation in silicon. Appl. Phys. Lett. 77 3749 (2000)
J.S. Williams, Y. Chen, J. Wong-Leung, A. Kerr, M.V. Swain: Ultra-micro-indentation of silicon and compound semiconductors with spherical indenters. J. Mater. Res. 14 2338 (1999)
L. Zhang, M. Mahdi: The plastic behavior of silicon subjected to micro-indentation. J. Mater. Sci. 31 5671 (1996)
A.E. Giannakopoulos, P.L. Larsson: Analysis of pyramid indentation of pressure-sensitive hard metals and ceramics. Mech. Mater. 25 1 (1997)
P.L. Larsson, A.E. Giannakopoulos: Tensile stresses and their implication to cracking at pyramid indentation of pressure-sensitive hard metals and ceramics. Mater. Sci. Eng., A 245 268 (1998)
ABAQUS Analysis User’s Manual Version 6.7 (SIMULIA, Providence, RI 2007)
J. Lee, G.L. Fenves: Plastic-damage model for cyclic loading of concrete structures. J. Eng. Mech. 124 892 (1998)
J. Lubiner, J. Oliver, S. Oller, E. Onate: A plastic-damage model for concrete. Int. J. Solids Struct. 25 299 (1989)
J. Lemaitre: A Course on Damage Mechanics 2nd ed (Springer, Berlin 1996)
W.C. Oliver, G.M. Pharr: An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments. J. Mater. Res. 7 1564 (1992)
H. Guckel: Silicon microsensors: Construction, design and performance. Microelectron. Eng. 15 387 (1991)
J.I. Jang, G.M. Pharr: Influence of indenter angle on cracking in Si and Ge during nanoindentation. Acta Mater. 56 4458 (2008)
Y.L. Tsai, J.J. Mecholsky: Fractal fracture of single crystal silicon. J. Mater. Res. 6 1248 (1991)
R. Ballarini, R.L. Mullen, Y. Yin, H. Kahn, S. Stemmer, A.H. Heuer: The fracture toughness of polysilicon microdevices: A first report. J. Mater. Res. 12 915 (1997)
A.M. Fitzgerald, R.H. Daukardt, T.W. Kenny: Fracture toughness and crack growth phenomena of plasma-etched single crystal silicon. Sens. Actuators, A 83 194 (1992)
M.Y. Tsai, C.H. Chen: Evaluation of test methods for silicon die strength. Microelectron. Reliab. 48 933 (2008)
L. Geng, Y. Shen, R.H. Wagoner: Anisotropic hardening equations derived from reverse-bend testing. Int. J. Plast. 18 743 (2002)
S. Shim, J. Jang, G.M. Pharr: Extraction of flow properties of single-crystal silicon carbide by nanoindentation and finite-element simulation. Acta Mater. 56 3824 (2008)
T.L. Anderson: Fracture Mechanics: Fundamentals and Applications 2nd ed (CRC Press, Boca Raton, FL 1995)
A. Hillerborgm, M. Modeer, P.E. Petersson: Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements. Cem. Concr. Res. 6 773 (1976)
A. Muchtar, L.C. Lim: Indentation fracture toughness of high purity submicron alumina. Acta Mater. 46 1683 (1998)
R.F. Cook, G.M. Pharr: Direct observation and analysis of indentation cracking in glasses and ceramics. J. Am. Ceram. Soc. 73 787 (1990)
S. Carlsson, S. Biwa, P-L. Larsson: On frictional effects at inelastic contact between spherical bodies. Int. J. Mech. Sci. 42 107 (2000)
J.T. Hagan: Cone cracks around Vickers indentation in fused silica glass. J. Mater. Sci. 14 462 (1979)
A. Kailer, Y.G. Gogotsi, K.G. Nickel: Phase transformation of silicon caused by contact loading. J. Appl. Phys. 81 3057 (1997)
K. Kese, D.J. Rowcliffe: Nanoindentation method for measuring residual stress in brittle materials. J. Am. Ceram. Soc. 86 811 (2003)
R. Tandon: A technique for measuring stresses in small spatial regions using cube-corner indentation: Application to tempered glass plates. J. Eur. Ceram. Soc. 27 2407 (2007)
P. Ostojic, R.M. Phersom: A review of indentation fracture theory: Its developments, principles and limitations. Int. J. Fract. 33 297 (1987)
R.D. Dukino, M.V. Swain: Comparative measurement of indentation fracture toughness with Berkovich and Vickers indenters. J. Am. Ceram. Soc. 75 3299 (1992)
G.M. Pharr: Measurement of mechanical properties by ultra-low load indentation. Mater. Sci. Eng., A 253 151 (1998)
D.B. Marshall, B.R. Lawn: Residual stress effects in sharp contact cracking: I. J. Mater. Sci. 14 200 (1979)
D.B. Marshall, B.R. Lawn: Residual stress effects in sharp contact cracking: II. J. Mater. Sci. 14 2225 (1979)
D.S. Harding: Cracking in brittle materials during low-load indentation and its relation to fracture toughness. Ph.D. Dissertation, Rice University, Houston, TX 1995
S. Wen, J. Bentley, J. Jang, G.M. Pharr: Cross sectional TEM studies of indentation-induced phase transformation in Si: Indenter angle effects, Fundamentals of Nanoindentation and Nanotribology IIIedited by K.J. Wahl, N. Huber, A.B. Mann,D.F. Bahr, and Y-T. Cheng (Mater. Res. Soc. Symp. Proc 841 Warrendale, PA 2005) R10.4
Y.Q. Wu, X.Y. Yang, Y.B. Xu: An HREM study of a lateral microcrack beneath indentation of [001] silicon. Acta Metall. Sinica 11 342 (1998)
A. Pajares, M. Chumakow, B.R. Lawn: Strength of silicon containing nanoscale flaws. J. Mater. Res. 19 657 (2004)
S-R. Jian: Mechanical deformation induced in Si and GaN under Berkovich nanoindentation. Nanoscale Res. Lett. 3 6 (2008)
J. Yan, H. Takahashi, X. Gai, H. Harada, J. Tamaki, T. Kuriyagawa: Load effects on the phase transformation of single-crystal silicon during nanoindentation tests. Mater. Sci. Eng., A 423 19 (2006)
S.J. Lloyd, J.M. Molina-Aldareguia, J.W. Clegg: Deformation under nanoindents in Si, Ge, and GaAs examined through transmission electron microscopy. J. Mater. Res. 16 3347 (2001)
J. Lankford, D.L. Davidson: The crack-initiation threshold in ceramic materials subject to elastic/plastic indentation. J. Mater. Sci. 14 1662 (1979)
T. Sata, K. Takamoto, H. Yoshikawa: Ultra-micro indentation hardness tester. Bull. Jap. Prec. Eng. 13 13 (1969)
T. Vodenitcharova, L.C. Zhang: A new constitutive model for the phase transformations in mono-crystalline silicon. Int. J. Solids Struct. 40 2989 (2003).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Wan, H., Shen, Y., Chen, Q. et al. A plastic damage model for finite element analysis of cracking of silicon under indentation. Journal of Materials Research 25, 2224–2237 (2010). https://doi.org/10.1557/jmr.2010.0270
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1557/jmr.2010.0270