On The Characterization of Plastic Flow in Zr based Metallic Glass Through Micro indentation: an Atomic Force Microscopy Analysis
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(2) 2618. T. Benameur, K. Hajlaoui, A. R. Yavari, A. Inoue and B. Rezgui. Fig. 1 AFM topographic image of 17.5 µm by 17.5 µm surface of Zr60 Ni10 Cu20 Al10 glass ribbon (shiny side) after indentation with 0.5 N charge and 15 s loading time.. Fig. 2 AFM lateral force image (deflection mode) of 40 µm by 40 µm area in Zr60 Ni10 Cu20 Al10 glass ribbon (dull side) after indentation with 2 N charge and 15 s loading time. Arrows indicate traces of circular flow fronts.. earlier.10, 11) Figure 3 shows AFM’s normal force image of typical overlapping bands in pile-up around indent after 2 N charge and 15 s loading time of the Zr60 Ni10 Cu20 Al10 glassy alloy. It is interesting to notice that in all impressions incomplete circular patterns are observed. Such configurations were swept by the silicon nitride tip in order to record their profiles. Figure 4(a) shows a cross section analysis of surface deformation features obtained after a 5 N loading charge. The three dimensional AFM topographic data show a nearly con-. Fig. 3 AFM normal force image (deflection mode) of overlapping bands in pile-up around indent on 12.5 µm by 12.5 µm region of Zr60 Ni10 Cu20 Al10 glass after indentation with 2 N charge and 15 s loading time.. Fig. 4 (a) Section analysis of AFM topographic data of glassy Zr60 Ni10 Cu20 Al10 after 5 N charge and 15 s loading time: a nearly consistent shear offset along the shear band (b) AFM image (deflection mode, 12.5 µm by 12.5 µm area) cross section of incomplete circular patterns in glassy Zr60 Ni10 Cu20 Al10 after 5 N charge and 15 s loading time.. sistent shear offset along the shear bands. The corresponding cross section of these circular patterns is seen in Fig. 4(b). The circular aspect of these bands is characterized by an almost irregular spacing varying from 1 to 2.5 µm as shown in Fig. 4(a). The pile-up observed around the indent is consistent with the macroscopic test response imposing the same stress state such as compressive7) or torsional.5) Moreover, their number increases with increasing load. Such behavior is consistent with the tendency for the metallic glass to deform in a strongly localized manner due to strain softening. The multiplication of shear bands is particularly true for metallic.
(3) On The Characterization of Plastic Flow in Zr-based Metallic Glass Through Micro-Indentation. glasses which present a perfect elastoplastic behavior during macroscopic stress-strain tests. 4. Discussion The plastic deformation behavior during the microindentation is compared to the response of metallic glass in a multiaxial stress state. The existence of the “pile-up” around indent could be correlated with macroscopic “stress-strain” response to uniaxial compressive tests where glassy metals present perfect elastoplastic behavior at room temperature. The recorded profile shown in Fig. 4(a), indicates clearly an average shear offset in the range of approximately 1 to 2.5 µm which is of the same order as that observed along a shear band in the Zr57 Ti5 Cu20 Ni8 Al10 BMG alloys (2 to 4 µm) under uniaxial compression.12) In recent work, the same multiple shear bands were observed around indents in glassy Zr41.25 Ti13.75 Cu12.5 Ni10 Be22.5 during instrumented sharp indentation, conducted with a maximum load of 10 N.3) However, the detailed three dimensional finite element simulation of instrumented indentation reported in previous investigations cannot predict multiple overlapping concentric rings in the pile-up, which are expected to be a combination of discrete and continuum phenomena. This work shows that the plastic deformation seems to be sensitive to shear bands’ nucleation and propagation mechanism as expected in geometries such as in bending or uniaxial compression. However, in other geometries such as uniaxial tension, the resulting deformation is unstable and brittle failure follows shortly after the onset of yielding.4, 5) While some researchers associate these variations to a mean stress effect2) and others to a dependence of the yield strength on the normal stress acting on the slip plane, described by a Mohr-Coulomb criterion,3) the numerical resolution of free volume equations developed below attempts to quantitatively outline the hydrostatic pressure on the emergence of multiple shear band in constrained geometries. The concentration of free volume is the order parameter adopted in the present work to explain the process of the appearance of multiple shear bands. Based on the free volume theory of Turnbull and Cohen,14–16) Steif et al.17) derived a flow equation for metallic glasses under a shear stress τ . Within the shear band, we assume that the metallic glass behaves homogeneously. Then, the shear strain rate is given by τ̇ α ∆G m τ ·Ω ∂γ = + 2 · f · exp − − sinh . ∂t µ ξ kB · T 2 · kB · T (1) where ∂γ ∂t is the constant strain rate, τ̇ is the rate of change of the applied stress, ξ is the concentration of free volume defined by ξ = vv∗f where vf the average free volume per atom and v ∗ a critical volume (≈ 0.8 ), α a geometrical factor, f the frequency of atomic vibration, ∆G m the activation energy, Ω the atomic volume, kB the Boltzmann’s constant, and T the absolute temperature. According to Spaepen,18) free volume is created by an applied shear stress τ and annihilated by a series of atomic jumps, the net rate of change of the free volume concentration, ξ , being given by:. 2619. Table 1 Typical properties used for the Zr60 Ni10 Cu20 Al10 glassy alloy. Material property (Gpa)19). Young’s modulus, E Poisson’s ratio, v 2) Shear modulus, µ (Gpa)19) Bulk modulus, B (Gpa)2) Thermal expansion coefficient, αth (◦ K)2) Glass transition temperature, Tg (◦ K)22) Average atomic volume, Ω (A3 ) Activation energy, ∆G m (J)20) Frequency of jumps, f (s−1 )20) Geometric factor, α 20) Constant of Boltzman, kB (J/◦ K). Value 96 0.36 35.3 114.3 10.1 × 10−6 625 16.4 10−19 1013 0.15–1 13.8 × 10−24. α· ∆G m ∂ξ = f · exp − − ∂t ξ kB · T 2α · kB · T 1 τ ·Ω · −1 − · cosh S · ξ · v∗ 2kB · T nD. (2). where n D is thenumberof atomic jumps required to annihilate v ∗ , and S = 23 1+v · µ , v is the Poisson’s ratio, and µ is the 1−v shear modulus. Equations (1) and (2) are solved numerically with the initial conditions τ (t = 0) = 0 and ξ(t = 0) ≈ 0.008.20, 21) The relevant material properties for the Zr60 Ni10 Cu20 Al10 glassy alloy used in calculations are given in Table 1. Figure 5(a) shows the shear stress-strain curve of the homogeneous deformation and Fig. 5(b) shows the concentration of free volume versus strain (at constant strain rate 0.02 s−1 ). The shear stress drops after an initial elastic response and the free volume concentration increases dramatically to facilitate the localization of strain in shear bands. For large strains, steady state stress and free volume are reached indicating a homogeneous softening of the metallic glass. The temperature effect is also illustrated in Figs. 5(a) and (b). Then, the homogeneous deformation reaches a steady state which depends on the temperature and the strain rate while the steady state values of the shear stress seem unaffected by a change of temperature at a constant strain rate, in contrast to the case of free volume concentration evolution. We note that the numerical solution is sensitive to several material parameters (such as ξ(t = 0), α, T , . . . etc.) which will be developed in separate work. The shear band emission behavior shown on experimental tests can be explained quantitatively by the free volume model. Under an applied shear stress τ , the free volume increases slightly in the competition between stressdriven creation of free volume and the process of diffusionassisted annihilation. Since certain amount of free volume exists, the increase becomes dramatic and leads to the nucleation of a local shear band accompanied by significant drop of the shear stress such as shown in Fig. 5(a). Therefore locally, equilibrium is established and stability is determined by the competition of the three processes: diffusion, annihilation and the stress-driven creation of free volume. Then, the same mechanism occurs in another zone which reaches the critical amount of free volume required to drop the shear stress. In some cases such as in uniaxial tensile tests, the localization of free volume is so severe that shear stress drops to a mini-.
(4) 2620. T. Benameur, K. Hajlaoui, A. R. Yavari, A. Inoue and B. Rezgui. Fig. 5 (a) Non-dimensionalized shear stress ξ -strain γ behavior of a metallic glass for a range of temperature. (b) Corresponding behaviour of the non-dimensionalized free volume concentration versus strain for the same range of temperature.. mum value before the localization stops and the deformation is concentrated into a unique shear band. Consequently, the metallic glass fails shortly after the onset of yielding without nucleation of shear bands. The important question now is about the reason why the strong localization of free volume in uniaxial tension does not occur in the compression. The recent study2) of the flow and fracture behavior of Zr– Ti–Ni–Cu–Be bulk metallic glass showed that the flow stress and fracture depend on the mean stress. Consequently, such deformation mechanisms would be expected to be pressuresensitive and depend on the hydrostatic component of the applied stress. While the mean stress effect is still under investigation, Flores et al.2) gave the following modification of the free volume model by describing the variation of initial free volume with mean stress, given by: ξ(t = 0) = ξ0 · 1 + σBm m for the compressive mean stresses and ξ(t = 0) = ξ0 · + Ω·σ B·v ∗ for tensile mean stresses where, ξ0 , is the initial free volume concentration with no superimposed mean stress and B is the bulk modulus. The resolution of eqs. (1) and (2) using different expressions for the initial free volume concentration evolution in the two cases of tensile and compressive mean. Fig. 6 (a) Non-dimensionalized shear stress ξ -strain γ behavior of a metallic glass for a range of a tensile mean stress. (b) Corresponding behavior of the free volume concentration versus strain for the same range of tensile mean stress.. stresses is shown in Figs. 6 and 7. A comparison between Figs. 6(a)–(b) and 7(a)–(b) shows how the stress versus strain curve is markedly altered by small superimposed tensile mean stress for a constant temperature and given strain rate, while no significant effect of compressive mean stress on free volume concentration is observed for the same range of mean stresses, which may explain the emergence of multiple shear bands prior to failure under localized indentation. 5. Conclusions AFM topographic measurements in contact mode were carried out for an analysis of multiple shear bands generated under micro indentation in Zr60 Ni10 Cu20 Al10 metallic glass under different loading charges, the experimental data revealed that: (1) Shear offset varies along shear band. Evidence for discrete displacement increments in the range of nanometers is observed. (2) The number of shear bands increases with increasing.
(5) On The Characterization of Plastic Flow in Zr-based Metallic Glass Through Micro-Indentation. 2621. ume model are used to explain quantitatively the reason for the generation of multiple shear bands under localized indentations. Acknowledgments This work was supported in part by the Rhône-Alpes grant Emergence 01 016759 02 and by the MES/DGRST technical facilities. REFERENCES. Fig. 7 (a) Nondimensionalized shear stress ξ -strain γ behavior of a metallic glass for a range of a compressive mean stress. (b) Corresponding behavior of the free volume concentration versus strain Where ξ0 = 0.05·, m ·Ω = Z and T = 400 K. α = 1, dγ /dt = 0.02 s−1 , σ2K T. loading in the range of 0.5 to 5 N. (3) Numerical calculations based on modified free vol-. 1) A. Inoue: Acta Mater. 48 (2000) 279–289. 2) K. M. Flores and R. H. Dauskart: Acta Mater. 49 (2001) 2527–2537. 3) R. Vaindyanathan, M. Dao, G. Ravichandran and S. Suresh: Acta Mater. 49 (2001) 3781–3789. 4) H. A. Bruck, A. J. Rosakis and W. L. Johnson: J. Mater. Res. 11 (1996), 503–508. 5) A. Bruck, T. Christman, J. Rosakis and W. L. Johnson: Scr. Metall. Mater. 30 (1994) 429–434. 6) C. J. Gilbert, V. Schroeder and R. O. Ritchie: Metall. Trans A 30 (1999) 1739–1753. 7) T. C. Hufnagel, P. El-Deiry and R. P. Vinci: Scr. Mater. 43 (2000) 1071– 1075. 8) F. Spaepen and Turnbull: Scr. Metall. 8 (1974) 563–568. 9) A. S. Argon: Acta Metall. 27 (1979) 47–58. 10) T. Benameur, A. Touhami and A. R. Yavari: J. Metastable Nanocryst. Mater. 8 (2000) 159–166. 11) T. Benameur and A. Touhami: Mater. Sci. Forum 386 (2002) 559–564. 12) A. El-Deiry, P. Vinci, N. Barbosa III and C. Hufnagel: Mat. Res. Soc. Symp. Proc. Vol 644 (2001). 13) P. Lowhaphandu, S. Montgomery and J. Lewandowski: Scr. Mater. 41 (1999) 19–24. 14) D. Turnbull and M. H. Cohen: J. Chemical Physics 34 (1961) 120–125. 15) D. Turnbull and M. H. Cohen: J. Chemical Physics 52 (1970) 3038– 3041. 16) M. H. Cohen and D. Turnbull: J. Chemical Physics 31 (1959) 1164– 1169. 17) P. S. Steif, F. Spaepen and J. W. Hutchinson: Acta Metall. 30 (1982) 447–455. 18) F. Spaepen: Acta. Metall. 25 (1977) 407–415. 19) A. Inoue: Materials Science Foundation, Vol. 6 (1999) pp. 1–35. 20) R. Huang, Z. Suo, J. H. Prevost and W. D. Nix: J. Mech. and Phys. of Solids 50 (2002) 1011–1027. 21) A. R. Yavari, P. Hicter and P. Desré: J. Chimie Physique 79 (1982) 579– 584 22) A. Inoue: Materials Science Foundation, Vol. 4 (1998) pp. 30–39..
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