handbook of materials modeling || bridging the gap between atomistics and structural engineering

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Perspective 14 BRIDGING THE GAP BETWEEN ATOMISTICS AND STRUCTURAL ENGINEERING J.S. Langer Department of Physics, University of California, Santa Barbara, CA 93106-9530, USA When Sid Yip asked me to write a commentary for this section of the handbook, I promptly reminded him that I am a co-author of a longer article in the section on mathematical methods. I told him that my article on amor- phous plasticity, written with Michael Falk and Leonid Pechenik, already is more of a departure from conventional ideas than may be appropriate for a book like this one, which should serve as a reliable reference for years into the future; and I asked whether I really ought to be given yet more space for expressing my opinions. Sid insisted that I should write the commentary any- way. So here are some remarks about one of the topics of interest in this book, the search for predictive models of deformation and failure of solids, and the role of nonequilibrium physics in this effort. Like many of my colleagues, I am impatient about the slow rate of progress in theoretical solid mechanics. We theorists have been given great opportu- nities. Remarkable developments in instrumentation and computation have advanced our knowledge about the atomic-scale behavior of solids far beyond what most of us could have imagined a decade or so ago; and yet it seems to me that our ability to bring that knowledge to bear on practical problems has not kept pace. I blame ourselves – the theorists – for this state of affairs. We have not been quick enough to explore new concepts that might move us from atomistic models and numerical simulations to engineering practice. To bridge this gap between atomistics and structural engineering, it seems almost trivially obvious that we need new phenomenologies. Our goal must be to develop predictive, quantitative and tractable descriptions of an enormously wide range of complex materials and processes. First-principles theories may be necessary to get us started, but they do not take us far enough by themselves, especially if they require that each physical situation be treated separately 2749 S. Yip (ed.), Handbook of Materials Modeling, 2749–2756. c 2005 Springer. Printed in the Netherlands.

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Page 1: Handbook of Materials Modeling || Bridging the Gap Between Atomistics and Structural Engineering

Perspective 14

BRIDGING THE GAP BETWEENATOMISTICS AND STRUCTURALENGINEERING

J.S. LangerDepartment of Physics, University of California, Santa Barbara, CA 93106-9530, USA

When Sid Yip asked me to write a commentary for this section of thehandbook, I promptly reminded him that I am a co-author of a longer articlein the section on mathematical methods. I told him that my article on amor-phous plasticity, written with Michael Falk and Leonid Pechenik, already ismore of a departure from conventional ideas than may be appropriate for abook like this one, which should serve as a reliable reference for years intothe future; and I asked whether I really ought to be given yet more space forexpressing my opinions. Sid insisted that I should write the commentary any-way. So here are some remarks about one of the topics of interest in this book,the search for predictive models of deformation and failure of solids, and therole of nonequilibrium physics in this effort.

Like many of my colleagues, I am impatient about the slow rate of progressin theoretical solid mechanics. We theorists have been given great opportu-nities. Remarkable developments in instrumentation and computation haveadvanced our knowledge about the atomic-scale behavior of solids far beyondwhat most of us could have imagined a decade or so ago; and yet it seems tome that our ability to bring that knowledge to bear on practical problems hasnot kept pace. I blame ourselves – the theorists – for this state of affairs. Wehave not been quick enough to explore new concepts that might move us fromatomistic models and numerical simulations to engineering practice.

To bridge this gap between atomistics and structural engineering, it seemsalmost trivially obvious that we need new phenomenologies. Our goal must beto develop predictive, quantitative and tractable descriptions of an enormouslywide range of complex materials and processes. First-principles theories maybe necessary to get us started, but they do not take us far enough by themselves,especially if they require that each physical situation be treated separately

2749S. Yip (ed.),Handbook of Materials Modeling, 2749–2756.c© 2005 Springer. Printed in the Netherlands.

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as in molecular dynamics simulations or even the most powerful multi-scaleanalyses. Moving across these length and time scales, learning how the small-scale phenomena fit together to produce complex, larger scale behaviors, isevery bit as important and challenging a research goal as is atomic-scaleinvestigation. Phenomenological research of the kind needed here requiresphysical insight to extract the general principles from the less relevant detailsand, therefore, it necessarily involves a substantial amount of guesswork.

The classic phenomenological models that are most relevant to deforma-tion and failure in solids are Hookean elasticity for static stress analysis and theNavier–Stokes equation for fluid dynamics. In both of those cases, the atomic-scale theories can in principle be used to compute constitutive parameters suchas elastic constants or viscosities; and those first-principles calculations arevery valuable in themselves because they tell us about the limits of validityof the phenomenological descriptions. Much of the value of phenomenology,however, lies in the fact that it is usually easier and more reliable to obtain theconstitutive parameters experimentally. The challenge is to make sure that thephenomenological framework truly captures the essential features of the sys-tems that we need to describe. In elasticity and fluid dynamics, the essentialingredients are Newton’s laws of motion plus continuity and symmetry crite-ria. Those must be the basic ingredients of a theory of solid plasticity, but theyare not sufficient.

The Navier–Stokes analogy is particularly relevant to my argumentbecause I want to talk mostly about noncrystalline materials. Deformable amor-phous solids are very similar to fluids in all but a few, albeit very important,respects. Although their molecular structures look very much like those of flu-ids, they support shear stresses, they exhibit stress-driven transitions betweenjammed and flowing states, and they even exhibit memory effects. Neverthe-less, because of their molecular-level similarities, I see no fundamental reasonwhy amorphous solids should not be amenable to a level of analysis roughlysimilar to that which we use for fluids. Moreover, I suspect that, once we havefound a useful way of describing the dynamics of deformation in amorphoussolids, we shall be well on our way to a useful description of polycrystallinematerials as well.

When I make remarks like these in public lectures, I am invariablyaccused (not always politely) of ignoring the huge body of literature and tra-dition in plasticity theory. Indeed, conventional approaches to plasticity havebeen extremely successful in conventional engineering applications; but manyof the problems that these theories must now confront – in biological materials,for example – are distinctly unconventional. Textbook treatments of plasticitygenerally appear in two different forms, one based on the usual formulation ofelasticity supplemented by phenomenological stress-strain relations and plas-tic yield criteria, and another liquid-like approach that focuses on rheologicalrelations between stresses and strain rates, usually with no reference to yield

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stresses or the irreversible deformations that occur in low-stress, non-flowingregimes. The hydrodynamic analogy, however, tells us that we should not try toseparate these two kinds of descriptions. Deformation or failure in a boundedmaterial under loading is usually a localized phenomenon; plastic flow occurswhere the stresses are large while, elsewhere, the stresses are relaxed and thematerial behaves nearly elastically. The material may harden in some placesand soften in others. To be truly useful, therefore, our phenomenological equa-tions of motion must incorporate all of those behaviors, and they must do soin a natural and relatively simple way.

What new concepts and analytic tools will be needed in order to develop aunified theory of this kind? How might those techniques differ from the oneswe have been using in the past? I shall try to start answering these questionsby pointing to some puzzles and internal inconsistencies that persist in theconventional theories. These puzzles include the question of how breakingstresses can penetrate plastic zones near crack tips, and the possibly relatedquestion of why brittle fracture becomes dynamically unstable at high speeds.I know of no convincing solution to either of those problems, certainly notfor the noncrystalline materials in which the definitive experiments have beenperformed; and the fact that these apparently simple problems have remainedunsolved for such a long time is, by itself, enough to convince me that there issomething seriously missing in our theories.

Here, however, I would like to focus on a few more basic questions that Ithink lead us to a better understanding of what the missing ingedients mightbe. The most elementary and familiar of these questions is: What are the fun-damental distinctions between brittle and ductile behaviors? A brittle solidbreaks when subjected to a large enough stress, whereas a ductile materialdeforms plastically. Remarkably, we do not yet have a deep understanding ofthe distinction between these two behaviors. Conventional theories of crys-talline solids say that dislocations form and move more easily through ductilematerials than brittle ones, thus allowing deformation to occur in one caseand fracture in the other. But the same behaviors occur in amorphous solids;thus the dislocation mechanism cannot be the essential ingredient of all theo-ries. Moreover, the brittleness or ductility of some materials depends upon thespeed of loading, which implies that a proper description of deformation andfracture must be dynamic; that is, it must be expressed in the form of equationsof motion rather than the conventional static or quasistatic formulations.

A second question that I find especially revealing is the following: Whatis the origin of memory effects in plasticity? Standard, hysteretic, stress-straincurves for deformable solids tell us that these materials – even the simplestamorphous ones – have rudimentary memories. For example, they “remem-ber” the direction in which they most recently have been deformed. Whenunloaded and then reloaded in the original direction, they harden and respondelastically, whereas, when loaded in the opposite direction, they deform

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plastically. The conventional way of dealing with such behavior is to specifyrules for how the response to an applied stress is determined by the history ofprior loading; but such rules provide little insight about the nature of a theorybased more directly on atomic mechanisms.

A much better way to deal with memory effects is to introduce internalstate variables that carry information about prior history and determine thecurrent response of the system to applied forces. The trick is to identify therelevant variables. I am coming to believe that this is one of the main pointsat which a gap opens between atomistic understanding and engineering prac-tice. All too often, for example, the plastic strain itself is used as such a statevariable. This procedure has its roots in the conventional Lagrangian formu-lation of solid mechanics in which deformations of a material are describedby displacements relative to fixed reference states. When applied to materialsundergoing irreversible plastic deformations, such a procedure violates basicprinciples of nonequilibrium physics because, if taken literally, it implies that amaterial somehow must remember its configurations at times arbitrarily far inthe past. That cannot be possible for an amorphous solid any more than it is fora liquid, where it is well understood that only displacement rates, and not thedisplacements themselves, may appear in equations of motion. When a solidundergoes a sequence of loadings and unloadings, bendings and stretchings,the displacement of an element of material from its original position cannotpossibly be a physically meaningful quantity, thus it cannot be a sensible wayof characterizing the internal state of the system. Nevertheless, the use of thetotal plastic strain, for example as a “hardening parameter,” appears frequentlyin the literature on plasticity.

What, then, are the appropriate state variables for amorphous solids? Myproposed answer to this question starts with the “flow-defect” or “shear-transformation-zone” (STZ) picture of Cohen et al. [1–4], in which plastic de-formation occurs only at localized sites where molecules undergo irreversiblerearrangements in response to applied stresses. Falk, Pechenik and I, in ourpaper in this volume (here denoted "FLP"), present a critical analysis of thoseearlier STZ theories, which I will not repeat in any detail here. On the plusside, these theories nicely satisfy my criteria for sensible phenomenologicalapproaches. Their central ingredient is an internal state variable, i.e., the den-sity of zones, and they generally postulate equations of motion for this density.On the other hand, they make a crucial assumption with which I disagree – thatthe plastic flow is equal simply to the density of zones multiplied by a stress-and temperature-dependent Eyring rate factor. The interesting behavior, then,is contained in various assumptions about rates of annihilation and creationof zones as functions of temperature and strain rate. Such theories can do rea-sonably well in accounting for some rheological and calorimetric behaviors of,say, metallic or polymeric glasses. They do not predict yield stresses, however,

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nor can they convincingly account for the wide range of dynamic behaviorobserved recently by Johnson and coworkers in bulk metallic glasses.

In the work described in FLP and in earlier papers [5–8], we have extendedand modified the original STZ theories in two ways. First, instead of assumingthat the STZ’s are structureless objects, we have modeled them as two-statesystems; that is, we have assumed that they transform back and forth betweentwo different orientations in response to applied stresses. This two-state pictureis inspired by molecular-dynamics simulations [5]. Among other implications,it tells us that we must supplement the scalar density by a tensorial quantitythat carries information about the orientations of the zones, and that the actualplastic flow will depend upon those orientations. The appearance of this in-ternal state variable produces entirely new dynamical properties, specifically,jamming behavior when the zones are all aligned with the stress and cannottransform further in the same direction, and a yield stress at which the jammedstate starts to flow.

Although the STZ’s are structural irregularities that live in an unperturbedmaterial for very long times, they are ephemeral in the sense that they arecreated and annihilated during irreversible deformations. These annihilationand creation terms play the same roles as those that appear in the original STZtheories. It is here that we have made the second of our basic changes usinga combination of phenomenological guesswork and the constraints imposedby symmetry and the second law of thermodynamics. In particular, we haveargued that the simplest possible creation rate is proportional to the rate atwhich energy is dissipated during plastic deformation, which is necessarily anon-negative scalar quantity. This phenomenological assumption leads us to arate factor that is substantially different from earlier versions, and which seemsfree from unphysical features.

As described in more detail in FLP, our equations of motion for the STZpopulations are best expressed in terms of two dimensionless state variables:�, a scalar field proportional to the density of STZ’s, and �i j , a tracelesssymmetric tensor field that describes the local orientation of the zones. The fulltheory is necessarily expressed in Eulerian coordinates, as in fluid dynamics.It consists of equations of motion for � and �i j supplemented by the usualacceleration equation relating the vector flow field vi to the divergence of thestress σi j , and an equation expressing the rate-of-deformation tensor as thesum of elastic and plastic parts. Because these equations refer to solids ratherthan liquids, they are necessarily more complicated than Navier–Stokes; butthey are capable of serving similar purposes.

The most unconventional result of this theory is the way in which theyield stress emerges. The equations of motion for an isotropic system havetwo kinds of steady state solutions at fixed applied stress. One of these solu-tions is jammed, i.e., non-flowing, and the other is unjammed. The jammed

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solution is dynamically stable below some stress (a function of the materialparameters) and is unstable above that stress. Conversely, the unjammedsolution is unstable below this “yield” stress and stable above it. Thus the con-ventional yielding criterion is replaced by an exchange of dynamic stabilitybetween two branches of steady-state solutions of a set of coupled, nonlineardifferential equations. The physical interpretation of this situation is that, atsmaller stresses, the two-state STZ’s become saturated in the direction of thestress – the magnitude of the orientational bias �i j reaches a stress-dependentmaximum – and the motion stops. At larger stresses, jammed zones are anni-hilated and unjammed ones created fast enough to sustain steady-state plasticflow.

The resulting dynamic version of an STZ theory reproduces a wide range ofthe phenomena observed in plastically deforming materials. Depending uponthe choice of just a small number of material parameters and initial conditions,theoretical sress-strain curves may exhibit work hardening, strain softening(for annealed samples with low initial densities of STZ’s), strain recovery fol-lowing unloading, Bauschinger effects, necking instabilities, and the like. Withthe addition of thermal fluctuations that cause spontaneous relaxation of theSTZ state variables, the theory quantitatively explains the experimentally ob-served transition between Newtonian viscosity at small loading to superplasticflow at larger stresses as a transition from thermally assisted creep at smallstress to plastic flow at the STZ yield stress.

There are also some interesting shortcomings. In the form described inFLP, the theory does not predict the results of calorimetric measurements.Also, like almost all other theories of plasticity, this version of the STZ theorylacks an intrinsic length scale. The theory does show signs of shear banding in-stabilities; but a complete theory of shear banding will have to predict both thewidth of the bands and the thickness of the transition region between flowingand jammed material.

I shall conclude my remarks by suggesting that these shortcomings may beassociated with a second theoretical gap in solid mechanics. I am thinking ofthe largely unexplored possibility that the statistical physics of a nonequilib-rium system such as a deforming solid, even when it is deforming very slowly,may be qualitatively different from that of a system in thermal and mechanicalequilibrium. Specifically, I want to raise the possibility that, during irreversibleplastic deformation, the slow, configurational degrees of freedom associatedwith molecular rearrangements may fall out of thermal equilibrium with thefast, vibrational degrees of freedom that couple strongly to a thermal reservoir.The statistical properties of both of these kinds of degrees of freedom may bedescribed by “temperatures”; and the two temperatures may be quite differ-ent from one another under nonequilibrium conditions. This kind of effectivetemperature is formally similar to the free volume introduced by Spaepen andothers, and can be used in much the same ways. The important difference is

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that the effective temperature is a measure of the configurational disorder inthe system. Like ordinary temperature, it is an intensive quantity, and does notnecessarily carry any implication of volume changes. It does, however, haveclear thermodynamic consequences.

Evidence for the existence of well defined effective disorder temperaturesis emerging from recent studies of granular materials, foams, and related sys-tems [9–13]. (So far, most of these studies are based on numerical simula-tions.) The elementary components of such systems, such as sand grains, aremuch too large for the ambient temperature to be relevant to their motions.Nevertheless, the use of fluctuation-dissipation relations in conjunction withmeasurements of diffusion constants, viscosities, stress fluctuations and thelike, yield estimates of effective temperatures that are nonzero and remarkablyconsistent with one another.

How might the addition of an effective disorder temperature resolve theremaining shortcomings of the STZ equations? In principle, this concept shouldbe closely related to the mechanism that we have postulated for the annihi-lation and creation of STZ’s. That is, the rate of energy dissipation associ-ated with plastic deformation must also be the heat source for the effectivetemperature. There also must be a cooling mechanism by which the effectivetemperature decreases in the absence of driving forces. The latter two effects,which couple the effective temperature to the ordinary bath temperature,should determine the calorimetric properties of the material.

Finally, it will be important that this disorder temperature diffuses fromhotter to cooler regions of the material; but its diffusion constant must naturallybe very much smaller than that for ordinary temperature because the associ-ated molecular rearrangements are very much slower than thermal vibrations.Higher effective disorder temperatures imply a higher density of STZ’s andthus a higher plastic strain rate at fixed stress, which in turn implies nonlin-ear amplification of the plastic response to driving forces. Preliminary inves-tigations indicate that the resulting theory predicts experimentally interestingbehavior of this kind. The effective thermal enhancement of plastic flow alsoappears to imply a picture of shear banding in which the flowing materialinside the band is “hotter” than the jammed material outside, and the thicknessof the boundary between the two regions is determined by the length scalecontained in the effective – not the ordinary thermal – diffusion constant.

In summary, I think that a dynamic version of the STZ theory has a goodchance of closing the gap between atomistics and engineering applications.Essential elements of the theory are the identification of physically meaningfulstate variables, the choice of rate factors that are consistent with basic princi-ples of nonequilibrium physics, and – perhaps – an effective disorder temper-ature to account for the fact that the configurational degrees of freedom mayfall out of equilibrium with the heat bath in systems undergoing irreversibledeformations.

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References

[1] D. Turnbull and M. Cohen, J. Chem. Phys., 52, 3038, 1970.[2] F. Spaepen, Acta Metall., 25(4), 407, 1977.[3] F. Spaepen and A. Taub, In: R. Balian and M. Kleman (ed.), Physics of Defects, Les

Houches Lectures, North Holland, Amsterdam, p. 133, 1981.[4] A.S. Argon, Acta Metall., 27, 47, 1979.[5] M.L. Falk and J.S. Langer, Phys. Rev. E, 57, 7192, 1998.[6] L.O. Eastgate, J.S. Langer, and L. Pechenik, Phys. Rev. Lett., 90, 045506, 2003.[7] J.S. Langer and L. Pechenik, Phys. Rev. E, 2003.[8] M.L. Falk, J.S. Langer, and L. Pechenik, Phys. Rev. E, 70, 011507, 2004.[9] I.K. Ono, C.S. O’Hern, D.J. Durian, S.A. Langer, A. Liu, and S.R. Nagel, Phys. Rev.

Lett., 89, 095703, 2002.[10] L. Cugliandolo, J. Kurchan, and L. Peliti, Phys. Rev. E, 55, 3898, 1997.[11] P. Sollich, F. Lequeux, P. Hebraud, and M. Cates, Phys. Rev. Lett., 78, 2020, 1997.[12] L. Berthier and J.-L. Barrat, Phys. Rev. Lett., 89, 095702, 2002.[13] D.J. Lacks, Phys. Rev. E, 66, 051202, 2002.