Dynamics of Large-Scale Plastic Deformation and the Necking Instability in Amorphous Solids

PHYSICAL REVIEW LETTERS week ending VOLUME 90, NUMBER 4 31 JANUARY 2003

L. O. Eastgate Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, New York 14853

J. S. Langer and L. Pechenik Department of Physics, University of California, Santa Barbara, California 93106 (Received 19 June 2002; published 31 January 2003) We use the shear transformation zone (STZ) theory of dynamic plasticity to study the necking instability in a two-dimensional strip of amorphous solid. Our Eulerian description of large-scale deformation allows us to follow the instability far into the nonlinear regime. We ﬁnd a strong rate dependence; the higher the applied strain rate, the further the strip extends before the onset of instability. The material hardens outside the necking region, but the description of plastic ﬂow within the neck is distinctly different from that of conventional time-independent theories of plasticity.

DOI: 10.1103/PhysRevLett.90.045506 PACS numbers: 62.20.Fe, 46.05.+b, 46.35.+z, 83.60.Df

Conventional descriptions of plastic deformation in when the strip is loaded slowly. One especially important solids consist of phenomenological rules of behavior, element of our analysis is our ability to interpret flow and with qualitative distinctions between time-independent hardening in terms of the internal STZ variables. and time-dependent properties, and sharply defined yield To make this problem as simple as possible, we consi- criteria. Plasticity, however, is an intrinsically dynamic der here only strictly two-dimensional, amorphous mate- phenomenon. Practical theories of plasticity should con- rials. By ‘‘strictly,’’ we mean that elastic and plastic sist not of intricate sets of rules, but of equations of displacement rates are separately planar as in two- motion for material velocities, stress fields, and other dimensional molecular dynamics simulations. The two- variables that might characterize internal states of solids. dimensional STZ equations presented in this paper are Roughly speaking, a theory of plasticity, especially for an based on earlier work by Falk, Langer, and Pechenik amorphous solid, should resemble the Navier-Stokes [7,11,12]. We use Eulerian coordinates in which, as in equation for a fluid, with the pressure replaced by a stress fluid dynamics, the variables xi denote the current physi- tensor, and the viscous forces replaced by a constitutive cal positions of material elements. Let the system lie law relating the rate of plastic deformation to the stresses in the x1 x, x2 y plane, and write the stress tensor 1 and internal state variables. That constitutive law should in the form: ij pij sij;p2 kk, where p is contain phenomenological constants, analogous to the the pressure and sij is the deviatoric stress — a trace- bulk and shear viscosities, that are measurable and, in less, symmetric tensor. In analogy to fluid dynamics, principle, computable from molecular theories. Yield cri- let vix; y; t denote the material velocity at the physi- teria, work hardening, hysteretic effects, and the like cal position x; y, and time t. Then the acceleration equa- would emerge naturally in such a formulation. tion is [13] The goal of the STZ (shear transformation zone) dvi @ij @p @sij theory of plasticity [1–7], from its inception, has been : (1) dt @x @x @x to carry out the above program. In this paper we show j i j how the STZ theory describes a special case of large- Here, is the density which, because we shall assume a scale yielding, specifically, the necking instability of a very small elastic compressibility and volume conserving strip of material subject to tensile loading. There is a large plasticity, we shall take to be a constant. The symbol d=dt literature on the necking problem. References that we denotes the material time derivative acting on a scalar or a have found particularly valuable include papers by vector field: Hutchinson and Neale [8], McMeeking and Rice [9], d @ @ and Tvergaard and Needleman [10]. Our purpose here is vk : (2) dt @t @x to explore possibilities for using the STZ theory to in- k vestigate a range of failure mechanisms in amorphous Our first main assumption is that the rate of deforma- solids, possibly including fracture. We are able to follow tion tensor can be written as the sum of linear elastic and the necking instability far into the nonlinear regime plastic contributions: where the neck appears to be approaching plastic failure 1 @v @vj D sij p while the outer regions of the strip become hardened and Dtotal i Dplast; ij 2 @x @x t 2 2K ij ij remain intact. We find that necking in the STZ theory is j i D rate dependent; the instability occurs at smaller strains (3)

045506-1 0031-9007=03=90(4)=045506(4)$20.00  2003 The American Physical Society 045506-1 PHYSICAL REVIEW LETTERS week ending VOLUME 90, NUMBER 4 31 JANUARY 2003 where is the shear modulus, K 1 =1 is yield stress.We also have assumed that the local density of the two-dimensional inverse compressibility (or bulk STZs is always at its equilibrium value so that we do not modulus), and is the two-dimensional Poisson ratio. need to solve an extra equation of motion for that field The symbol D=Dt denotes the material time derivative (denoted by the symbol in earlier papers). acting on any tensor, say Aij: The important exception alluded to above is the pres- ence of the absolute-value bars in Eq. (7). The expression DAij @Aij @Aij vk Aik !kj !ik Akj; (4) inside the bars is proportional to the rate at which plastic Dt @t @xk work is being done on the system, a quantity which appears in the original theory as a non-negative factor and !ij is the spin: in the STZ annihilation and creation rates. A negative 1 @vi @vj value of this quantity would be unphysical. In earlier !ij : (5) 2 @xj @xi studies of spatially uniform systems, this quantity always remained positive; however, we have observed negative The plastic part of the rate-of-deformation Dplast,like ij values in the present calculations. The absolute value s , is a traceless symmetric tensor, thus the plastic defor- ij prevents such unphysical behavior and is consistent with mations are area conserving. For present purposes, we use the intent of the original theory.We emphasize, however, a simple, quasilinear form of the STZ theory in which that this term contains some of the principal assumptions plast of the STZ theory. There are other possibilities for it (see, Dij 0 qijs; ; qijs; sij ij; (6) for example, [2]); and it will be interesting to explore the and 0 is a material-specific constant. The traceless, sym- physical significance of these variations of the model. metric tensor ij is the internal state variable mentioned To understand the transition between viscoelastic and earlier. It is proportional to a director matrix that specifies viscoplastic behaviors at the yield stress, and the role the orientation of the STZs; its magnitude is a measure played by the state variable , it is easiest to look of the degree of their alignment. The equation of motion first at a uniform system under pure shear. Let sxx for ij is syy s, sxy 0, xx yy , xy 0; and consider a situation in which s is held constant. Equations (3) D 1 ij q jq s j : (7) and (7) become Dt ij 2 km km ij "_ s ; (8) In Eq. (6), plays—very roughly—the role of the ‘‘back 0 stress’’ or ‘‘hardening’’ parameter in conventional theo- _ s 1 s; (9) ries of plasticity [14–16], a major difference being that emerges directly from a rate equation governing the where "_ is the total strain rate. At s 1, these equations population of STZs and is, in principle, a directly mea- exhibit an exchange of stability between the nonflowing surable quantity [1,17]. If the second term on the right- steady-state solution with "_ 0, s for s<1, and the hand side of Eq. (7) were missing, then would be flowing solution with "_ Þ 0, 1=s for s>1.Asex- proportional to the integrated plastic strain. This second plained in earlier publications, the steady-state system is term, however, which is produced by the creation and ‘‘jammed’’ or ‘‘hardened’’ in the direction of the applied annihilation of STZs, is a crucial element of the STZ stress for s<1; whereas, for s>1, new STZs are being theory. As we shall show briefly below, this term produces created as fast as existing ones transform, and there is a the exchange of dynamic stability between viscoelastic nonzero plastic strain rate. and viscoplastic states that replaces the conventional as- Our goal now is to see how this exchange of stability sumptions of yield surfaces and other purely phenomeno- occurs in a dynamic, spatially nonuniform situation. logical rules of behavior. Consider a rectangle with straight grips at x Lt. With one important exception, Eqs. (6) and (7) con- The upper and lower surfaces, at y Yx; t, are free stitute a tensorial version of the original STZ theory boundaries. We assume symmetry about both the x and y obtained by linearizing the stress dependence of the axes so that we need to consider only the first quadrant of rate factors and rescaling. Because of the linearization, the system. On the free upper boundary, the relation these equations do not properly describe memory effects between the material velocities and the motion of the present in the full theory that are important when the surface is system is unloaded or reloaded, but this will not affect @Y @Y our results until the system reaches the necking instabil- vyx; Y; tvxx; Y; t : (10) ity. Only after the neck starts to flow plastically, causing @t x @x t the hardened regions to unload, will we need the full We also must specify stress conditions on this surface: nonlinear theory to determine if the observed behavior Y00 is pertinent. We have chosen the rescaling so that all nn ; nt 0: (11) stresses and moduli are expressed in units of the plastic 1 Y023=2

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Here, is the surface tension, is the curvature, and the subscripts n and t denote normal and tangential compo- a nents, respectively. The grips at x Lt move out- ward at a predetermined strain rate, LL=L_ ; thus b vxL; y; tL for 0

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1.05 like fracture, although necking differs from ordinary fracture in that the stress concentration that triggers the s instability is due to narrowing of the strip as a whole 1 ∆ rather than to a localized defect on just one surface. ∆ Nevertheless, the behaviors shown in Fig. 3 (and other, later-stage results not shown here) suggest the onset of a

s and 0.95 localized, propagating failure mechanism. In order to study the connection between necking and fracture in adequate detail we believe that we shall need to use the 0.9 full STZ theory and to improve our numerical resolution. 024 This research was supported primarily by U.S. De- x−axis partment of Energy Grant No. DE-FG03-99ER45762. It was also supported in part by the MRSEC Program of FIG. 3. The deviatoric stress s,and, along the x axis. This the NSF under Award No. DMR96-32716 and by a grant picture corresponds to case (a) indicated by the big circle in from the Keck Foundation for Interdisciplinary Research Fig. 2 shortly after the sample is starting to neck. in Seismology and Materials Science. Lance Eastgate has been supported by the Research Council of Norway. flow anywhere until the stress exceeds the yield stress in We thank L. Petzold, R. McMeeking, A. Needleman, and the necking region. In the opposite limit, when the load- J. Rice for useful advice. ing is fast, remains appreciably smaller than s for a longer time during which the material undergoes plastic deformation everywhere. It would be useful to test this prediction of the STZ theory by measuring necking, say, *Present address: Department of Physics, University of California, Santa Barbara, CA 93106. in amorphous metals. We presume that various ingre- [1] M. L. Falk and J. S. Langer, Phys. Rev. E 57, 7192 (1998). dients of the full STZ theory, such as stress-dependent [2] A. E. Lobkovsky and J. S. Langer, Phys. Rev. E 58, 1568 rate factors and other features that have been ignored (1998). here, would be needed to fit experimental data quantita- [3] M. L. Falk, Phys. Rev. B 60, 7062 (1999). tively, and that we would learn much about the theory [4] J. S. Langer and A. E. Lobkovsky, Phys. Rev. E 60, 6978 from such an effort. (1999). Note that the behavior shown in Fig. 3 is quite different [5] M. L. Falk and J. S. Langer, MRS Bull. 25, 40 (2000). from that predicted by conventional, time-independent [6] J. S. Langer, Phys. Rev. E 62, 1351 (2000). plasticity theory, in which there would be a plastic zone [7] J. S. Langer, Phys. Rev. E 64, 011504 (2001). with s 1 inside the neck, that is, s would remain at the [8] J.W. Hutchinson and K.W. Neale, Acta Metall. 25, 839 yield stress. We have found no evidence that this conven- (1977). [9] R. M. McMeeking and J. R. Rice, Int. J. Solids Struct. 11, tional behavior occurs in the simulations presented here, 601 (1975). even for the smallest pulling speeds. (For cavitation, the [10] V. Tvergaard, A. Needleman, and K. K. Lo, J. Mech. STZ theory predicts a conventional plastic zone around a Phys. Solids 29, 115 (1981). growing hole when the growth rate is very slow [4].) Once [11] M. Falk, Ph.D. thesis, UCSB, 1998. the instability sets in, the development of the neck is [12] L. Pechenik, D. J. Rabinowitz, and J. S. Langer, Bull. governed by the elastic energy already stored in the strip. Am. Phys. Soc. 46, 1023 (2001). We have confirmed this feature of late-stage necking [13] L. E. Malvern, Introduction to the Mechanics of a Con- dynamics by performing numerical experiments in which tinuous Medium (Prentice-Hall, Inc., New Jersey, 1969). we stop the motion of the grips, that is, hold them fixed, at [14] J. Lubliner, Plasticity Theory (Macmillan Publishing various times after the neck has started to form but well Company, New York, 1990). before it has grown appreciably.We find that so long as the [15] Unified Constitutive Laws of Plastic Deformation, edited by A. S. Krausz and K. Krausz (Academic Press, stored elastic energy is large enough, stopping the remote San Diego, 1996), Chap. 6, p. 281. loading in this way has almost no effect on the neck; it [16] J. Aboudi and S. R. Bodner, Int. J. Eng. Sci. 18, 801 (1980). continues to grow just as before, driven by the elastic [17] G. Gagnon, J. Patton, and D. J. Lacks, Phys. Rev. E 64, unloading. 051508 (2001). The behavior described in the last sentence — necking [18] P.N. Brown, A. C. Hindmarsh, and L. R. Petzold, SIAM driven by stored elastic energy — looks in many ways J. Sci. Comput. 15, 1467 (1994).

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