Modeling the Dynamic Propagation of Shear Bands in Bulk Metallic Glasses B.J. Edwards, K. Feigl, M.L. Morrison*, B. Yang*, P.K. Liaw*, and R.A. Buchanan* Dept. of Chemical Engineering, The University of Tennessee, Knoxville, TN 37996 Dept. of Mathematical Sciences, Michigan Tech. University, Houghton, MI 49931 *Dept. of Materials Science and Engineering, The University of Tennessee, Knoxville, TN 37996 Keywords: shear bands, micromechanical modeling, bulk amorphous materials, finite element analysis, tension test Abstract: A model is developed for quantifying shear-band propagation in bulk metallic glasses. This model is written in terms of the stress, temperature, and band propagation speed. The model quantifies the shear-band length, width, and speed, as well as the direction of propagation and the magnitude of displacement across the band. Corresponding author: B.J. Edwards, bjedwards@chem.engr.utk.edu (865) 975-9596
The formation of shear bands in materials under an applied stress has been studied extensively over the past two decades; however, truly dynamic models of the propagation of these bands are still in the process of being developed. Recent efforts have made remarkable strides toward understanding this phenomenon, but continuum models that can describe the essential physics of shear band propagation are still largely unavailable. In the present work, a macroscopic model is presented for describing the propagation and dissipation of shear bands in bulk metallic glasses (BMGs). This model is written in terms of three relevant variable fields, the stress tensor, σ (Pa), the absolute temperature, T (K), and the flux (propagation speed) of the free volume, φ (m/s), as well as experimentally determined parameters, such as the mass density, heat capacity, conductivity, etc. Finite-element (FE) calculations of the resulting model equations allow the estimation of the characteristic features of the shear band, such as the width, length, and speed of propagation, as well as the temperature, stress, and displacement profiles along the band length. In a recent article [1], the evolution of shear bands in a Zr-based BMG was observed in-situ during tensile loading via thermographic imaging with a high-speed infrared (IR) camera. In these experiments [1], Zr-based BMG Zr 52.5 Cu 17.9 Ni 14.6 Al 10.0 Ti 5.0 samples were subjected to tensile loading in a load-control mode with a loading rate of 44.5 N/s. An IR camera recorded thermographic images of the sample surface at a frame rate of 725 Hz, which amounted to one camera frame every 0.00138 s. After a linear stress-strain regime at low loads (with Young s modulus, E 0, of about 100 GPa [2,3]), plastic deformation began and the onset of shear-band generation occurred. The heat generated during the shear deformation along the band was measured 2
using the IR camera. The experiment was continued to the ultimate sample failure, and up to this point, shear bands continually originated, propagated, and arrested. The IR camera measured the thermal trace of the shear bands after propagation had been arrested. It is important to realize that the thermal trace of the shear band can be significantly longer and wider than the shear band itself. It is apparent that the propagation velocity of the shear bands is very fast relative to the IR camera frame rate. Since the IR camera rate records images at intervals of 0.00138 s, very few, if any, shear bands were actually captured during propagation; most images were taken of shear bands at their full lengths after the propagation had been arrested. Although the width of a typical shear band in a BMG is estimated at about 10-50 nm, the width of the thermal trace left behind by the band decreases from about 0.4 mm at the initiation site to a smaller value at the shear-band tip. The temperature decrease along the length of the thermal trace is roughly 2 K [1]. The temperature near the initiation site of a shear band is estimated to be on the order of 800 K [4]; however, this is confined to a region of width on the order of the 50 nm. According to scanning-electron microscopy images of the sample taken after the completion of the experiment, all shear bands observed had roughly the same characteristics [1]. Each band propagated in one of two directions, at an angle of either 56 or 124 taken counterclockwise relative to the long axis of the sample, and the width was roughly 30 nm. Also, the deformation of transverse planes across the shear band decreased along the band length from the origination site to the tip. In the previous paper [1], a free-volume exhaustion mechanism on the atomiclength scale was proposed to explain the shear-band evolution [4]. When a shear band 3
initiates, a large amount of free volume is created at the origination site by the localized strain and temperature profiles. As the shear band propagates, the free volume in the plastic deformation regime will be swept along with the shear band tip, until the local strain and temperature profiles can no longer produce enough of a free-volume increase to allow further propagation. Thus, the band arrests as the available free volume exhausts itself due to decreasing levels of strain and temperature along the length of the band. In this letter, a continuum model is presented for describing quantitatively the macroscopic dynamics of the shear-band propagation and arrest. This model was developed using the principles of non-equilibrium thermodynamics, as represented through a mathematical framework [5,6]. This framework allows one to derive coupled evolution equations (for a set of macroscopic variables), which are guaranteed to satisfy the laws of thermodynamics. Therefore, very complicated interactions occurring among different physical phenomena can be described in a consistent fashion. As mentioned above, the variable set used in this formulation is composed of σ, T, and φ, each representing a function of both space and time. The coupled evolution equations for these variables were derived as described above, and found to be σ σ κ σ κ + 1 σ = η κ t λ T ( + κ ) T T : :, (1) T T ρcˆ V = σ : κ + ( k T ) + ρ Q ( Λ φt ), (2) t φ Q = φ + Λ T t λ φ. (3) The first equation represents the evolution in space and time of the stress distribution within the sample. The form of this equation is that of the classical Upper- 4
Convected Maxwell Model of viscoelasticity [5]. In the absence of shear-band formation, the stress distribution can be determined from the experimental stress-strain curve. In the experiment under investigation herein [1], the stress evolution was 11 = 4.4395 10 t 4.171 10 7 7 quantified by the equation σ ( ) + along the long axis of the sample., where the 1-direction lies T In Eq. (1), κ represents the velocity gradient tensor. In the absence of shearband formation, the imposed velocity gradient is a uniaxial extensional field. The elongational strain rate, ε&, follows also from the experimental stress-strain curve [1]: it takes the value 0.0022 s -1 for t t [0,28], and follows the equation ε& = ( 8.25 10 4 ) e 0. 037 for t (28,37]. (Note that the onset of the plastic deformation regime occurs at 28 s.) The viscosity is related to λ (s) through the shear modulus [5], η( Pa s) = Gλ, herein taken as G = 32.3 GPa for the BMG sample used in the experiment [7]. The bulk sample viscosity at the onset of the plastic deformation regime was estimated from the experiment as on the order of 10 11 Pa s at ambient temperature [1]. This parameter is 11 represented by an Arhennius temperature dependence: this function is taken as η = 10 Pa s for T T g and > = ( 10 19 ) exp( 69 T ) η T g for T g T [8], where T = 675 K [8-10]. Equation (2) is the evolution equation for the temperature field. The first three terms in this expression comprise the standard temperature equation in fluid and solid continuum mechanics. The first term on the right hand-side (RHS) is a thermoelastic coupling between temperature and stress, and the second term on the RHS accounts for heat conduction. The additional fourth term in Eq. (2) quantifies the degree of coupling occurring between the temperature evolution and the shear-band propagation. This term represents the effect of the flux of free volume on the temperature distribution within the g 5
sample. A similar coupling term appears in Eq. (3). Of the parameters appearing in this equation, the mass density, ρ = 6. 73 g/cm 3 [3,7,10], the specific heat, C ˆ = 24 J/(g-atom K) [96], and the thermal conductivity, k = 4 W/(m K) [3], are all taken as constants. Q is a dimensionless parameter arising from the definition of the free energy of the material; herein, it is assigned the value of unity so that it drops out of the analysis. The last parameter, Λ (m 2 /s 2 K), is an anisotropic matrix that quantifies the extent of coupling between the free-volume propagation and temperature evolution. In general, the elements of this matrix are functions of T, φ, and the invariants of the stress tensor; however, at present, these elements are taken as linear functions of the applied load. The final expression, Eq. (3), describes the evolution of the shear band in terms of its speed of propagation, φ. The RHS describes the dissipative dynamics of the shearband propagation. The first term on the RHS represents the relaxation of the driving force for the shear-band propagation. This rate of relaxation is governed by the V relaxation time, λ Q (s). It is, in general, a function of temperature, but will be taken φ as a constant herein. The second term on the RHS represents the coupled effect that the temperature distribution has on the flux of free volume within the sample. Initial indications of the model behavior were obtained computationally. Simulations were performed using the 2-d version of Eqs. (1)-(3) on a rectangular grid of sample dimensions. The FE method was used for the spatial discretization, and a Runge- Kutta method was used for the temporal discretization. The simulated experiment begins at the time t = 0. The temperature is assumed uniform throughout the sample domain at 292 K at t = 0. For t [0,28], the deformation 6
is elastic, and no shear banding is observed. Boundary conditions on the stress are provided from the known applied load. For t (28,37], shear bands initiate, propagate, and arrest, starting from the edge of the sample. At the macroscopic level, shear band initiation must be implemented directly since there is no description of microstructural defects built into the model. Here, for a shear band initiating at time t (28,37], a temperature spike, T, was 0 imposed in a 50 50 nm 2 region near the right sample edge. As already mentioned, T 0 should be on the gross order of 1,000 K. For t > t, Eqs. (1)-(3) were solved and the results compared to experimental data [1]. Parameter values used are T0 = 1,000 K, 4 λ = 1.5 s, Λ = Λ = Λ = γ = 1.25 Load (GPa) 1. 875 m 2 /s 2 K [5], and φ 10 11 ( + ( tan 2 56 1) / tan 56 ) Λ 22 = γ 1. 12 21 Figures 1 and 2 compare model results to experimental data taken for a shear band that originated at a load of 1.62 GPa. IR thermographic images were taken immediately following the initiation of the shear band. The temperature decreases with distance along the shear band, and with distance perpendicular to it. The dimensions of the thermal trace left behind after the shear band has propagated and arrested can, thus, be determined as functions of time. The model captures all of the qualitative features of the data, and provides reasonable quantitative fits as well. The magnitude of φ is graphed in Figure 3 as a function of the distance along the shear band at various times. The shear band accelerates rapidly at first, obtaining its maximum velocity soon after initiation at a short distance from the point of origin. From this point in time, the maximum velocity relaxes as its location in space is translated 7
along the length of the shear band. Simultaneously with the above process, the maximum temperature is pulled along the length of the shear band, thus distributing the applied temperature increase away from the origination site and along the full length of the band. Intense stress softening occurs in the region of the shear band tip. Note from Figure 3 that the shear band propagates and arrests well within the time span of one camera frame. Thus, only its thermal trace is left behind for subsequent camera frames to capture. The length of the shear band as estimated from Figure 3 is roughly 0.35 mm, which is much less that the length of the thermal trace evident in Figure 1. By measuring the distance perpendicular to the band (at a specified location along it) for which the maximum temperature (with respect to time) is above T, an estimate can be made of the width of the shear band. For the model parameters specified above, the width of the shear band is in the range of 40-80 nm at the initiation site, and this value decreases g moving away from this location. The width of φ perpendicular to the band might also be used to estimate the band width. Figure 4 illustrates that the model predicts an approximately linear increase in the length of the shear bands with the applied load, until very near the final fracture point of the experiment for temperatures in the range of 800-1,200 K. Near this value of the load, the shear-band length begins to increase dramatically; this occurrence is due to two factors. First, the shear band length calculated from the solution of Eqs. (1)-(3) increases faster than linearly at higher loads. Second, the thermoelastic term on the RHS of Eq. (2) starts to influence the system significantly at higher loads, essentially acting as a heat source in the evolution equation for the temperature; the thin, continuous line in Figure 4 shows the effect of removing the thermoelastic contribution. Note that varying T for a 0 8
given value of the applied load also affects the shear-band length, possibly accounting for the scatter in the experimental data: the various shear-band-activation events occur at different initial temperatures. However, the range of lengths calculated from the model for the temperature range of 800 1,000 K is actually fairly narrow: as T increases, the propagation speed increases as well. Consequently, the driving force behind the shearband propagation dissipates quicker as well, thus limiting the potential size increase of the shear band. Figure 5 displays a plot of the displacement across the shear band as measured experimentally using SEM, and as predicted from the model. Displacement is calculated with the model by integrating the φ y element with respect to time at various points along the length of the band. This results in a virtual translation, which one could identify with the transverse displacement perpendicular to the band. According to Figure 5, most of the displacement occurs during the first 75% of the band length. Afterward, the displacement drops dramatically as the band tip is approached. Interestingly, the model predicts that a negative displacement always occurs for bands oriented at 56, and a positive displacement always occurs for bands oriented at 124. 0 References [1] Morrison ML, Yang B, Liu CT, Liaw PK, Buchanan RA, Carmichael CA, Leon RV. Phys Rev Lett, submitted for publication. [2] Bian Z, He G, Chen GL. Scripta Mater 2000;43:1003. [3] Aydmer CC, Üstündag E, Hanan JC. Metall Mater Trans 2001;32A:2709. [4] Steif PS, Spaepen F, Hutchinson JW. Acta Metall 1982;30:447. 9
[5] Beris AN, Edwards BJ. Thermodynamics of Flowing Systems. Oxford University Press: New York; 1994. [6] Grmela M, Öttinger HC. Phys Rev E 1997;56:6620. [7] Bian Z, Pan MX, Zhang Y, Wang WH. Appl Phys Lett 2002;81:4739. [8] Mukherjee S, Schroers J, Zhou Z, Johnson WL, Rhim W-K. Acta Mater 2004;52:3689. [9] Glade SC, Busch R, Lee DS, Johnson WL, Wunderlich RK, Fecht HJ. J Appl Phys 2000;87:7242. [10] Mattern N, Kühn U, Hermann H, Roth S, Vinzelberg H, Eckert J. Mat Sci Eng A 2004;375:351. Figures Figure 1: Temperature relative to ambient versus distance along the shear band from the point of initiation. Results are displayed at three camera frames after shear-band propagation. Figure 2: Temperature relative to ambient versus distance perpendicular to the shear band for the same case as Figure 1. Figure 3: The magnitude of φ versus distance along the shear band from the point of origin at various times. Figure 4: Shear-band length versus applied load for three values of initial temperature. Figure 5: Displacement at various locations along the shear band. 10
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Figure 1, Edwards et al. 12
Figure 2, Edwards et al. 13
Figure 3, Edwards et al. 14
Figure 4, Edwards et al. 15
Figure 5, Edwards et al. 16