A hybrid atomistic-continuum finite element modelling of nanoindentation and experimental verification for copper crystal

Paweł Dłużewski; Marcin Maździarz; Grzegorz Jurczak; Piotr Traczykowski; Koichi Niihara; Roman Nowak; Krzysztof Kurzydłowski

Paweł Dłużewski Institute of Fundamental Technological Research
Marcin Maździarz Institute of Fundamental Technological Research
Grzegorz Jurczak Institute of Fundamental Technological Research
Piotr Traczykowski Institute of Plasma Physics and Laser Microfusion
Koichi Niihara Nagaoka University of Technology
Roman Nowak Nordic Hysitron Laboratory, Helsinki University of Technology
Krzysztof Kurzydłowski Warsaw University of Technology

Abstract

Problem of locally disordered atomic structure is solved by using a hybrid formulation in which nonlinear elastic finite elements are linked with discrete atomic interaction elements. The continuum approach uses nonlinear hyperelasticity based upon the generalized strain while the atomistic approach employs the Tight-Binding Second-Moment Approximation potential to create new type of elements. The molecular interactions yielding from constitutive models of TB-SMA were turned into interactions between nodes to solve a boundary value problem by means of finite element solver. In this paper we present a novel way of modelling materials behaviour where both discrete (molecular dynamics) and continuum (nonlinear finite element) methods are used. As an example, the nanoindentation of a copper sample is modelled numerically by applying a hybrid formulation. Here, the central area of the sample subject to nanoindentation process is discretised by an atomic net where the remaining area of the sample far from indenters tip is discretised by the use of a nonlinear finite element mesh.

Keywords

References

[1] M.E. Bachlechner, A. Omeltchenko, A. Nakano, R.K Kalia, P. Vashishta, I. Ebbsjo, A. Madhukar. Dislocation emission at the silicon/silicon nitride interface: A million atom molecular dynamics simulation on parallel computers. Phys. Rev. Lett., 84: 322,2000.
[2] J.Q. Broughton, F.F. Abraham, N. Berstern, E. Kaxiras. Concurrent coupling of length scales: Methodology and application. Phys. Rev. B, 60: 2391-2403, 1999.
[3] D. Chrobak, K Nordlund and R. Nowak. Non-dislocation origin of GaAs nanoindentation pop-in event. Phys. Rev. Lett., 98: 045502, 2007.
[4] F. Cieri, V. Rosato. Tight-binding potentials for transition metals and alloys. Phys. Rev. B, 48: 22-33, 1993.
[5] P. Dłużewski, P. Traczykowski. Numerical simulation of atomic positions in quantum dot by means of molecular statics. Archives of Mechanics, 55: 501-515, 2003.
[6] P. Dłużewski. Anisotropic hyperelasticity based upon general strain measure. Computational Material Science, 29: 379, 2004.
[7] P. Dłużewski, G. Maciejewski, G. Jurczak, S. Kret, J.-y' Laval. Nonlinear FE analysis of residual stresses induced by dislocations in heterostructures. J. Elasticity, 60: 119-129, 2000.
[8] M.J. Horodon, B.L. Averbach. Precision density measurements on deformed copper and aluminum single crystals. Acta Metallurgica, 9: 247, 1961.
[9] Y.R. Jeng, C.M. Tan. Theoretical study of dislocation emission around a nanoindentation using a static atomistic model. Phys. Rev. B, 69: 104109, 2004.
[10] G. Jurczak, G. Maciejewski, S. Kret, P. Dluzewski, P. Ruterana. Modelling of indium rich clusters in MOCVD InGaN/GaN multilayers. Journal of Alloys and Compounds, 382: 10-16,2004.
[11] J. Knap, M. Ortiz. An analysis of the quasicontinuum method. J. Mech. Phys. Solids, 49: 1899-1923, 200l.
[12] E. Lidorikis, M.E. Bachlechner, R.K Kalia, A. Nakano, P. Vashishta, G.Z. Voyiadjis. Coupling length scales for multiscale atomistic-continuum simulations: Atomistically-induced stress distributions in Si/Si3N4 nanopixels. Phys. Rev. Lett., 87: 086104, 200l.
[13] Y. Liu, B. Wang, M. Yoshino, S. Roy, H. Lu, R. Komanduri. Combined Numerical Simulation and Nanoindentation for Determining Mechanical Properties of Single Crystal Copper at Mesoscale. J. Mech. Phys. Solids, 53: 2718-2741, 2005.
[14] R. Nowak, T. Manninen, K Heiskanen, T. Sekino, A. Hikasa, K Niihara, T. Takagi. Peculiar surface deformation of sapphire: Numerical simulation of nanoindentation. Appl. Phys. Lett., 83: 5214-5216, 2003.
[15] R. Nowak, F. Yoshida, D. Chrobak, KJ. Kurzydłowski, T. Takagi, T. Sasaki. Nanoindentation examination of crystalline solid surfaces in Encyclopedia of Nanoscience and Nanotechnology. Am. Sci. Publ. ed. S.H. Nalwa, 2008, in press.
[16] R.W. Ogden. Non-linear Elastic Deformations. Ellis Horwood, Chichester, 1984.
[17] H. Rafii-Tabar, L. Hua, M. Gross. A multi-scale atomistic-continuum modelling of crack propagation in a twodimensional
macroscopic plate. J. Phys. Condens. Matter, 10: 2375-2387, 1998.
[18] V.B. Shenoy, R. Miller, E.B. Tadmor, R. Phillips, M. Ortiz. Quasicontinuum models of interfacial structure and deformation. Phys. Rev. Lett., 80: 742-745, 1998.
[19] V.B. Shenoy, R. Miller, E.B. Tadmor, D. Rodney, R. Phillips, M. Ortiz. An adaptive finite element approach to atomic-scale mechanics - the quasicontinuum method. J. Mech. Phys. Solids, 47: 611-642, 1999.
[20] A. Smirnowa, L.V. Zhigilei, B.J. Garrison. A combined molecular dynamics and finite element method technique applied to laser induced pressure wave propagation. Comput. Phys. Commun., 118: 11- 16, 1999.
[21] F. Spaepen. Interfaces and stresses in thin films. Acta Materialia, 48: 31, 2000.
[22] E.B. Tadmor , R. Philips, M. Ortiz. Mixed atomistic and continuum models of deformation in solids. Langmuir, 12: 4529-4534, 1996.
[23] E.B. Tadmor, M. Ortiz, R. Philips. Quasicontinuum analysis of defects in solids. Phil. Mag. A, 73: 1529, 1996.
[24] C. Teodosiu. Elastic Models of Crystal Defects. Springer-Verlag and Editura Academiei, Berlin-Bucuresti, 1982.
[25] S.N. Vaidya, G.C. Kennedy. Compressibility of 18 elemental solids to 45 kb. J. Phys. Chem. Solids, 31: 2329-2345, 1970.
[26] S.N. Vaidya, G.C. Kennedy. Compressibility of 22 elemental solids to 45 kb. J. Phys. Chem. Solids, 33: 1377-1389, 1972.
[27] E. Weinan , Z. Huang. A dynamic atomistic-continuum method for the simulation of crystalline materials. J. Comput. Phys., 182: 234-261, 2002 .