[1]
V. Yamakov, D. Wolf, S. R. Phillpot, H. Gleiter, Grain-boundary diffusion creep in nanocrystalline palladium by molecular-dynamics simulation., Acta Materialia 50 (2002) 61-73.
DOI: 10.1016/s1359-6454(01)00329-9
Google Scholar
[2]
V. Yamakov, D. Wolf, S. R. Phillpot, H. Gleiter, Deformation mechanism crossover and mechanical behaviour in nanocrystalline materials., Philosophical Magazine Letters 83 (2003) 385-393.
DOI: 10.1080/09500830031000120891
Google Scholar
[3]
M. Buehler, A. Hartmaier, H. Gao, Atomistic and continuum studies of crack-like diffusion wedges and dislocations in submicron thin films, will appear in Journal of the Mechanics and Physics of Solids, 2003.
DOI: 10.1016/j.jmps.2003.09.024
Google Scholar
[4]
M. Buehler, A. Hartmaier, H. Gao, Atomistic and continuum studies of diffusional creep and associated dislocation mechanisms in thin films on substrates, in: K. Hemker, D. Lassila, L. Levine, H. Zbib (Eds.), MRS Symposium Proceedings 779, Materials Research Society, Pittsburgh, USA, 2003, p. W4.7.1.
DOI: 10.1557/proc-779-w4.7
Google Scholar
[5]
J. Schiøtz, T. Vegge, F. D. Tolla, K. Jacobsen, Atomic-scale simulations of the mechanical deformation of nanocrystalline metals, Physical Review B-Condensed Matter 60 (1999) 11971.
DOI: 10.1103/physrevb.60.11971
Google Scholar
[6]
H. van Swygenhoven, P. Derlet, Grain-boundary sliding in nanocrystalline fcc metals, Physical Review B-Condensed Matter 64 (2001) 224105.
DOI: 10.1103/physrevb.64.224105
Google Scholar
[7]
A. Hasnaoui, H. van Swygenhoven, P. Derlet, Dimples on nanocrystalline fracture surfaces as evidence for shear plane formation, Science 300 (2003) 1550-1552.
DOI: 10.1126/science.1084284
Google Scholar
[8]
R. P. Vinci, E. M. Zielinski, J. C. Bravman, Thin Solid Films 262 (1995) 142.
Google Scholar
[9]
M. J. Kobrinsky, C. V. Thompson, The thickness dependence of the flow stress of capped and uncapped polycrystalline Ag thin films, Applied Physics Letters 73 (1998) 2429-2431.
DOI: 10.1063/1.122471
Google Scholar
[10]
R.-M. Keller, S. P. Baker, E. Arzt, Acta Materialia 47 (1999) 415.
Google Scholar
[11]
L. B. Freund, The stability of a dislocation in a strained layer on a substrate, Journal of Applied Mechanics 54 (1987) 553-557.
DOI: 10.1115/1.3173068
Google Scholar
[12]
W. D. Nix, Mechanical properties of thin films, Metallurgical Transactions A 20 (1989) 2217-2245.
Google Scholar
[13]
W. D. Nix, Yielding and strain hardening of thin metal films on substrates, Scripta Materialia 39 (2001) 545-554.
DOI: 10.1016/s1359-6462(98)00195-x
Google Scholar
[14]
B. von Blanckenhagen, P. Gumbsch, E. Arzt, Dislocation sources in discrete dislocation simulations of thin film plasticity and the Hall-Petch relation, Modelling and Simulation in Matererials Science & Engineering 9 (2001) 157-169.
DOI: 10.1088/0965-0393/9/3/303
Google Scholar
[15]
G. Dehm, T. Balk, B. von Blanckenhagen, P. Gumbsch, E. Arzt, Dislocation dynamics in sub-micron confinement: recent progress in Cu thin film plasticity, Zeitschrift f¨ur Metallkunde 93 (2002) 383-391.
DOI: 10.3139/146.020383
Google Scholar
[16]
M. Legros, K. Hemker, A. Gouldstone, S. Suresh, R.-M. Keller-Flaig, E. Arzt, Microstructural evolution in passivated Al films on Si substrates 15during thermal cycling, Acta Materialia 50 (2002) 3435-3452.
DOI: 10.1016/s1359-6454(02)00157-x
Google Scholar
[17]
B. von Blanckenhagen, P. Gumbsch, E. Arzt, Dislocation sources and the flow stress of polycrystalline thin metal films, Philosophical Magazine Letters 83 (2003) 1-8.
DOI: 10.1080/0950083021000050287
Google Scholar
[18]
T. Balk, G. Dehm, E. Arzt, Parallel glide: unexpected dislocation motion parallel to the substrate in ultrathin copper films, Acta Materialia 51 (2003) 4471-4485.
DOI: 10.1016/s1359-6454(03)00282-9
Google Scholar
[19]
H. Gao, L. Zhang, W. D. Nix, C. V. Thompson, E. Arzt, Crack-like grainboundary diffusion wedges in thin metal films, Acta Materialia 47 (1999) 2865-2878.
DOI: 10.1016/s1359-6454(99)00178-0
Google Scholar
[20]
K. Schwarz, Simulation of dislocations on the mesoscopic scale. I. methods and examples, II. application to strained-layer relaxation, Journal of Applied Physics 85 (1999) 108-129.
DOI: 10.1063/1.369430
Google Scholar
[21]
L. Nicola, E. Van der Giessen, A. Needleman, Discrete dislocation analysis of size effects in thin films, Journal of Applied Physics 93 (2003) 5920-5928.
DOI: 10.1063/1.1566471
Google Scholar
[22]
I.-H. Lin, R. Thomson, Cleavage, dislocation emission, and shielding for cracks under general loading, Acta Metallurgica 34 (1986) 187-206.
DOI: 10.1016/0001-6160(86)90191-4
Google Scholar
[23]
T. Mura, The Continuum Theory of Dislocations, in: H. Herman (Ed.), Advances in Materials Research, Vol. 3, Interscience Publishers, New York, 1968, Ch. 1.
Google Scholar
[24]
H. H. M. Cleveringa, E. van der Giessen, A. Needleman, Comparison of discrete dislocation and continuum plasticity predictions for a composite material, Acta Materialia 45 (1997) 3163-3179.
DOI: 10.1016/s1359-6454(97)00011-6
Google Scholar
[25]
M. C. Fivel, T. J. Gosling, G. R. Canova, Implementing image stresses in a 3D dislocation simulation, Modelling and Simulation in Matererials Science & Engineering 4(6) (1996) 581-596.
DOI: 10.1088/0965-0393/4/6/003
Google Scholar
[26]
A. Hartmaier, M. C. Fivel, G. R. Canova, P. Gumbsch, Image stresses in a free-standing thin film, Modelling and Simulation in Matererials Science & Engineering 7 (1999) 781-793.
DOI: 10.1088/0965-0393/7/5/310
Google Scholar
[27]
M. Y. Gutkin, A. E. Romanov, Straight edge dislocations in a thin twophase plate, physica status solidi 125 (1991) 107-125.
DOI: 10.1002/pssa.2211250108
Google Scholar
[28]
J. P. Hirth, J. Lothe, Theory of Dislocations, reprinted (second) Edition, Krieger Publishing Company, Malabar, Florida, 1992.
Google Scholar
[29]
L. P. Kubin, G. Canova, M. Condat, B. Devincre, V. Pontikis, Y. Br´echet, Dislocation microstructures and plastic flow: A 3D simulation, Solid State Phenomena 23&24 (1992) 455-472.
DOI: 10.4028/www.scientific.net/ssp.23-24.455
Google Scholar
[30]
I. Kaur, W. Gust, L. Kozma, in: Handbook of Grain and Interface Boundary Diffusion Data, Vol. 1, Ziegler Press, Stuttgart, Germany, 1989, pp.380-390.
Google Scholar
[31]
M. Buehler, A. Hartmaier, H. Gao, Hierarchical multi-scale modeling of plasticity of submicron thin metal films, submmitted to Modelling Simulation Mater. Sci. Eng., 2003. 166 Appendix The tensor Gijk as it is employed in this work to calculate the stress field of a dislocation in an infinite bi-material is listed here. The formulation has been adapted from the works of Mura [23] and Gutkin and Romanov [27]. G111 = D � 6(x2 − d) r2 − − 4(x2 − d)3 r4 − − (5A + B)x2 +9(A + B)d r2 + + 4A(x2 + d)(3d2 +10dx2 + x2 2) r4 + − 32Adx2(x2 + d)3 r6 + � G112 = D� − 2x1 r2 − + 4x1(x2 − d)2 r4 − + (3A − B)x1 r2 + + 4Ax1(3d2 +4dx2 − x2 2) r4 + − 32Adx1x2(x2 + d)2 r6 + � G221 = D� − 2(x2 − d) r2 − + 4(x2 − d)3 r4 − + (A + B)x2 − (3A − B)d r2 + + 4A(x2 + d)(d2 − 6dx2 − x2 2) r4 + + 32Adx2(x2 + d)3 r6 + � G222 = D� − 2x1 r2 − − 4x1(x2 − d)2 r4 − + (A + B)x1 r2 + + 4Ax1(d2 + x2 2) r4 + + 32Adx1x2(x2 + d)2 r6 + � G121 = D� − 2x1 r2 − + 4x1(x2 − d)2 r4 − + (A + B)x1 r2 + − 4Ax1(d2 +4dx2 + x2 2) r4 + + 32Adx1x2(x2 + d)2 r6 + � 17G122 = D� − 2(x2 − d) r2 − + 4(x2 − d)3 r4 − + (3A − B)x2 − (A + B)d r2 + + 4A(x2 + d)(d2 +6dx2 − x2 2) r4 + − 32Adx2(x2 + d)3 r6 + � G211 = G121 ,G212 = G122 The constant D is defined as D = µ1/[4π (1 − ν1)], with µ1 and ν1 being shear modulus and Poisson ratio of that part of the bi-material containing dislocations. The other part of the bi-material is regarded as dislocation-free in our treatment. The variable d denotes the distance to the bi-material interface. The following abbreviations are used: A =(1− Γ)/(1 + k1Γ) ,B=(k2 − k1Γ)/(k2 +Γ) Γ=µ2/µ1 ,ki =3− 4νi ,i=1, 2 r2 ± = x2 1 +(x2 ± d)2 . 18E (GPa) νb(nm) B (Pa s) η 114.0 0.33 0.26 10−4 100 Table 1 Material parameters employed in the discrete dislocation simulations, where E is the Young's modulus, ν the Poisson ratio, b the norm of Burgers vector, and B the phonon drag coefficient. These values mimic material properties of copper. The ratio between glide and climb mobility is denoted by η. 192x 1x σ0 1 ν, 1E f 2 h , 2νE Fig. 1. Geometry of a thin film with thickness hf on an infinite substrate. Film and substrate can have different elastic moduli E and ν. Under and applied tensile stress σ0 dislocations with Burgers vector parallel to the x1-axis climb down the grain boundary (black) which causes nucleation of parallel glide dislocations (grey) of the same Burgers vector. s∆ , ,ν11 2 h b f 2ν E E b Fig. 2. The layer of virtual boundary dislocations with Burgers vectors ˆb on the thin film geometry as shown in Figure 1. The surface is discretized into segments of length ∆s. The schematic is drawn for a dislocation configuration with arbitrary Burgers vectors b, represented here by a single dislocation. 2002 04 06 0 t (ns) 0 1 2 3 σ11 (GPa) A C B σinf 0 100 200 300 400 x1 (nm) 0 50 100 150 x2 (nm) A 0 100 200 300 400 x1 (nm) 0 50 100 150 x2 (nm) B 0 100 200 300 400 x1 (nm) 0 50 100 150 x2 (nm) C Fig. 3. Relaxation of the applied stress by diffusional creep to a constant level σinf (top). The three inserts at the bottom show the dislocation configuration at the times indicated in the stress-time diagram. 210 1 02 03 0 dsrc (nm) 1.0 1.2 1.4 1.6 1.8 2.0 σinf (GPa) hf = 30nm hf = 150nm hf = 300 nm Fig. 4. The flow stress σinf, after relaxation by grain boundary diffusion without parallel glide dislocations as a function of the distance of the nucleation site from the film surface. Three different film thicknesses as given in the legend have been investigated. 0 10203040 1µm / h 0.0 0.5 1.0 1.5 2.0 σinf (GPa) (a) dsrc = 2.6 nm (a) dsrc = 0.1 h (b) dsrc = 2.6 nm (b) dsrc = 0.1 h Fig. 5. Flow stress versus inverse film thickness for different climb dislocation nucleation criteria and grain boundary treatments, as given in the legend: (a) stands for absorbing grain boundaries, and (b) for blocking grain boundaries. Different locations dsrc of the climb dislocation sources have been investigated. 221 10 100 1000 η 0.00 0.25 0.50 0.75 1.00 σinf (GPa) Fig. 6. The influence of the ratio η of glide and climb mobilities on the flow stress of a film of thickness hf = 30 nm with a dislocation source at dsrc =0.26 nm is practically negligible. 23
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