arXiv:cond-mat/0610641v1 [cond-mat.dis-nn] 23 Oct 2006 hcsadsi ytm:peitosfo hoyo contin of theory a from predictions systems: slip and Shocks rmti eocl esetv?Hr edsusthree. discuss we Here perspective? emerge mesoscale naturally this predictions from and properties formulation. other formation continuum What Wall this temper- from low naturally crystals. emerges at deformed thus formed plastically walls in high cell atures at the formed and boundaries grain temperatures the sponta- continuum both the a of structures explanation are dislocations—providing wall-like by shocks the formed these neously of that origin argued We fundamental shock developed [2]. laws both singularities climb, that suppress numerically to showed law ap- and modified closure a a and proposed dynamics proximation, microscopic the rederived from appro- subsequently law We (hence this climb temperatures). and high at glide dislocations priate both of allowing density the crystals, for in law recently grain evolution [1] of an Roy evolution and proposed Acharya and formation polycrystals. in the boundaries as well as crystals, soitdsrs losi oda nadasr neigh- absorb and in wall’s draw the dislocations. to how dislo- boring it about allows and and stress stress walls, associated the the near of density predictions behavior cation detailed asymptotic makes grain the It of about walls. formation cell the and with boundaries associated stresses residual oesbl upsnuaiyi h ilcto density. dislocation the form in will singularity jump systems subtle these more that active. a is predicts system model slip the one only Instead, where plasticity) in I structures stage such tal experimen- of and not simulations lack will (discrete-dislocation the systems dislocation explaining of walls, kind cell one form only predicts with model materials the climb, that forbid to restricted propriately rydniyde aeacs singularity. cusp en- a The have components). does stress density (despite individual ergy boundary the grain in a jumps across the resid- continuous the is with stress associated ual energy elastic the that predicts (1) of deformation plastic the governs motion Dislocation (2) ewl eieteetrepeitosaayial by analytically predictions three these derive will We (3) ilcto est up nsingle-slip in jumps density Dislocation up neeg density. energy in Cusps eiulstress. Residual ASnmes 46.,91.60.Dc,91.60.Ed,47.40.-x dens earl numbers: dislocation In PACS the (3) in singularities density. plastici jump energy I of stage elastic bound formation (single-slip the active the grain is in At dislocation cusp of (2) a type be process. will de formation there we grain-boundary dislocations; the neighboring w of attracting cell by plasticity-induced crys boundary and in the boundaries formation grain boundary across grain jump and plasticity about tions sn eetydvlpdcniumter fdislocation of theory continuum developed recently a Using h otnu oe predicts model continuum The aoaoyo tmcadSldSaePyis lr Hall, Clark Physics, State Solid and Atomic of Laboratory h otnu model continuum The uaht Limkumnerd Surachate onl nvriy taa Y18320,USA 14853-2501, NY Ithaca, University, Cornell hnap- When . Dtd uy1,2018) 14, July (Dated: ∗ n ae .Sethna P. James and n o’ eiaino h qain fmto estudy. we motion of equations the Acharya of and derivation notation Roy’s predic- our and striking introducing by these begin We modify and tions. affect the likely by will ignored mechanisms model physical contin- how are the and of predictions model, structure uum Finally, detailed these the whether upon depend predictions. and to likely how these consider out will flesh we to simulations dimen- cal one in onto theory sion dynamics dislocation the mapping u hc esrstentflxo dislocation of flux net the measures which ersnsteeatc eesbedsoto n h plastic tensor the distortion and distortion reversible elastic, the represents est esr[,4 ,6 7] 6, 5, dislocation 4, Nye net [3, the the tensor by of density described result dislocations, the of is density distortion plastic The formation. t oho of borhood ie ntrso function a of terms in given urn rmasnl ilcto oigwt velocity with J moving dislocation single a from current ilcto rvn t oinis motion its driving dislocation a where ety n hnmd lsr prxmto owrite indepen- to approximation move closure to a mo- J dislocation made of then each equation and allowed the dently, we of [2], derivation tion our In dislocations). eutb ipyasmn htaltedsoain move velocity dislocations same the all the that with assuming simply by result ie h e urn o h osre ugr etrof vector tensor, Burgers density conserved the dislocation for the current net the gives ie h oc density force the times ihBresvector Burgers with , ij y,cl al ilntfr;isedw predict we instead form; not will walls cell ty), opeemcocpcdsrpino h deformation the of description macroscopic complete A h ieeouino h lsi itrintno is tensor distortion plastic the of evolution time The famtra sgvnby given is material a of ntrsof terms in = ε ity. σ ls hc efcnitnl cst form to acts self-consistently which alls, iln ieteaypoi aetm dynamics late-time asymptotic the rive as 1 hr ilb eiulstress residual a be will There (1) tals. stelclsrs de o xml,t h other the to example, for (due, stress local the is yais edrv he e predic- new three derive we dynamics, tgso lsiiy hnol one only when plasticity, of stages y t ρ l ij re omda ihtemperatures, high at formed aries ugr equation Burgers b x j ( v . x ρ n = ) δ o n cay 1 o h aefinal same the got [1] Acharya and Roy . ( β ξ ij ,adtentPahKelrfreon force Peach–Koehler net the and ), − P ε ecie h reesbepatcde- plastic irreversible the describes u ilcto dynamics dislocation uum ilm † b v F ∂ nte(oregand neigh- (coarse-grained) the in , ie yacntn ( constant a by given , l l β = mj P ewl rsn numeri- present will We . ∂ J i − ij u = ∂ j ε t = lmn ρ X = ij α f ∂ l β PK ρ t = t β mc ij E i α ij P − b = + σ j α where , ε nc δ ilm β − ( α ξ ij P ε ntelocal the on agn to tangent , α ∂ where , lmn (1) ) l J mj D ε t m ilm ( The . b ρ c ) J v σ / β mj nc 2) ij is E 2 dislocations: We now specialize to the case of λ = 0, where glide and climb are treated on an equal footing (applicable to grain RA D D Jij = εial lρaj = εial(εlmnρmcσnc)ρaj boundary formation during polygonization at high tem- 2 F − 2 (2) peratures, for example). In this case, equation 6 tells us D P = (σicρac σacρic)ρaj . that the individual components of β are all independent 2 − of one another, slaves solely to the evolution of the total A physically natural choice for D(ρ) is proportional to stress energy density . By contracting equation 6 with ¯ P E an inverse density of dislocation lines (so that the force Cijkmβkm then using expression 7, the time evolution of per dislocation drives the motion [1]); we choose here to the strain energy becomes discuss the mathematically more convenient choice of a 1 2 ∂t + (∂z ) =0 . (8) constant D. (Numerically, both give qualitatively similar E 2 E evolution [2].) We controlled the microscopic mobility Equation 8 can be cast into the famous Burgers equation difference between glide and climb [2] by introducing a by defining = ∂z [8, 9, 10]: mesoscale parameter λ. By setting F E ∂t + ∂z =0. (9) F F F P RA λ RA ∂ β = J = J δ J , (3) The scalar (z) is again the net Peach–Koehler force t ij ij ij 3 ij kk − density on theF local dislocation density ρ(z). Burgers at low temperatures λ = 1 removed the trace of J enforc- equation is the archetype of hyperbolic partial differen- ing volume conservation, and hence forbids climb, while tial equations. Under Burgers equation will develop F at high temperatures λ = 0 allowed for equal mobilities sharp jumps downward after a finite evolved time, corre- for both glide and climb. sponding to cusps in the energy density and leading to P E We now turn to an analysis of the continuum equation jumps in the components of β (Fig 1). in one dimension. Near a wall singularity (say, perpen- dicular to ˆz) the dynamics is one-dimensional. The varia- tions of the stress, plastic strain, and dislocation densities parallel to the wall asymptotically become unimportant F compared to the variations along ˆz as one approaches the singularity. In one dimension the stress σ(x), generally given by a long-range integral over the neighboring dis- locations ρ(x′), can be written as a linear function of the local plastic distortion βP:

P E σij = C¯ijkmβ (4) − km where 2ν C¯ijkm = µ δ¯ikδ¯jm + δ¯imδ¯jk + δ¯ij δ¯km (5)  1 ν  − P PSfrag replacements βxz (not quite equal to the elasticity tensor), and the Kro- necker delta is modified such that δ¯zz = 0. Hence the evo- P P lution law for β simplifies dramatically. The βzj com- P z ponents do not evolve (except βzz for λ = 0, which helps enforce volume conservation); the other6 components all evolve according to FIG. 1: Cusps and jumps in one dimension. For the continuum dynamics equation allowing both glide and climb, F P 1 P λ P the Peach–Koehler force density obeys Burgers equation, ∂tβij = (∂z ) ∂z βij βkkδij (6) and hence develops sharp jumps (top). The individual com- P −2 E  − 3  ponents of the plastic distortion tensor β (bottom), as well as the stress and strain tensors, evolve to also have sharp where we have rescaled the time to set D = 1. The elastic jumps at the walls; the dislocation density hence develops the energy density in equation 6 can be written in terms E δ-function singularity associated with grain boundary forma- of the local plastic distortion tensor: tion. These discontinuities in the residual stresses and strains, however, cancel out in the net elastic energy density, which is 1 P P µ P P 2 continuous with only a cusp at the walls (middle). = C¯ijkmβ β = (β + β ) E 2 ij km 2 xy yx 2 2 µν + µ βP + βP + (βP + βP )2 (7) At late times, the asymptotic late-time solutions to xx yy 1 ν xx yy Burgers equation (between the singularities) are linear   − 3 functions of z whose slopes decay with time:

z z0 − (10) F ∼ t t0 − The corresponding elastic energy density (continuous across the singularities) has asymptotic form given by integrating equation 10:

2 1 (z z0) − + 0 (11) E ∼ 2 (t t0) E − The individual components of the distortion tensor (apart from the three time-independent components βzj ) numerically take the form

P z z0 βij αij − + γij , (12) ∼ √t t0 − which can be shown to be consistent with the evolution law for the energy density (equation 11), so long as the FIG. 2: Cusps formed with one slip system. Plastic dis- coefficients αij and γij obey the relations P tortion tensor βyx formed by climb-free evolution of a Gaus- sian random initial state of edge dislocations pointing along ˆz C¯ijkmαij αkm =1 , C¯ijkmαij γkm =0 , (13) with Burgers vector along ±xˆ. Notice that walls do not form C¯ijkmγij γkm =2 0 , with one slip system, only cusps in the distortion tensor; com- E pare to [21]. for i, j = x or y; there are no restrictions on αiz and γiz. There has been extensive work on discrete point dis- location simulations in two dimensions, where it appears Which of these new predictions of the mesoscale dis- necessary to include more than one slip system to form location dynamics model can be trusted? Which seem walls [11, 12, 13, 14, 15, 16, 17, 18]. What does the (in retrospect) physically plausible, and which are likely continuum model predict for this case? Let us con- artifacts of simplifications made in the modeling process, sider a 2D system (constant along zˆ) of straight edge or just mathematical curiosities of this particular model? dislocations with Burgers vector along xˆ, described by Residual stress. a single non-zero component ρzx(x, y). Such a system (1) The primary driving force for dis- has two non-zero components of the distortion tensor, location motion is stress. In cases where dislocations P P dynamically assemble into walls (polygonization at high ρzx = ∂xβyx + ∂yβxx. A simulation− of this system with Gaussian random ini- temperatures, cell wall formation at low temperatures), P P it does seem natural in retrospect to expect that the tial βyx and βxx, allowing both glide and climb, generates a series of walls of dislocations roughly parallel to they ˆ- walls will be associated with stress jumps designed to axis similar to those seen by Barts and Carlsson [19] in attract residual dislocations to the walls. Grain bound- their 2D study of single slip with both glide and climb. aries formed from the melt when separate growing crys- (See figure 3.) tals touch should likely not be described by the contin- To forbid climb in this case it is convenient [22] to uum model [23]. These residual stress jumps must be P viewed as a fundamental prediction of the model. simply choose βxx 0. Figure 2 shows the evolution of the distortion field≡ for climb-free dynamics with a single (2) Cusps in the energy density. If there are jumps in slip system. In agreement with experiment and the dis- the stress at grain boundaries, surely it is natural that crete dislocation simulations [20, 21], we observe no cell there be some singularity in the energy density. At late wall structures in single slip (which would correspond to stages when the dislocations between grain boundaries jumps in βP in Fig 2). Instead, we find a network of have all been removed, a flat boundary can lower the sys- surfaces exhibiting a striking new singularity: a cusp in tem energy by moving into the region of higher energy the distortion tensor, corresponding to a jump in the dis- density. If both glide and climb are allowed [24], and if location density. We can understand this singularity an- dislocation mobility is unimpeded (by precipitates, im- alytically using our mapping to Burgers equation. With purities, lattice pinning, or tangling) this traction will only one non-zero component of βP, the energy density lead the boundaries to move until the energy density is P is proportional to the square of βyx (eqn 7). The energy continuous across the boundary. Hence, for mobile walls density satisfies eqn 8 and forms cusps, so the distortion at high temperatures, it is natural to expect the energy tensor must also form cusps. density to be continuous, and have only cusp singulari- 4

jump structure in plastically deformed systems with only one active slip system.

We thank Amit Acharya for helpful conversations, and acknowledge funding from NSF grants ITR/ASP ACI0085969 and DMR-0218475.

∗ Electronic address: [email protected] † URL: http://www.lassp.cornell.edu/sethna/sethna.html [1] A. Roy and A. Acharya, Journal of the Mechanics and Physics of Solids 53, 143 (2005). [2] S. Limkumnerd and J. P. Sethna, Phys. Rev. Letters 96, 095503 (2006). [3] J. F. Nye, Act. Metall. 1, 153 (1953). [4] J. D. Eshelby (Academic Press, San Diego, 1956), pp. 79–144. [5] A. M. Kosevich, Sov. Phys. JETP 15, 108 (1962). [6] E. Kr¨oner, Kintinuumstheorie der Versetzungen und FIG. 3: Continuum of walls. The dislocation density tensor Eigenspannungen (Springer Verlag, Berlin, 1958). ρzx evolved allowing both glide and climb from a random [7] T. Mura, Phil. Mag. 8, 843 (1963). initial state of edge dislocations along t ∼ ˆz with b ∼ xˆ. [8] U. Frisch and J. Bec, in Les Houches 2000: New Trends Notice that the dislocations arrange themselves into small- in Turbulence, edited by M. Lesieur, A.Yaglom, and angle tilt boundaries at the lattice scale, but do not coarsen; F. David (Springer EDP-Sciences, Berlin, 2001), pp. 341– compare to [19]. 83. [9] J. P. Sethna, M. Rauscher, and J.-P. Bouchaud, Euro- physics Letters 65, 665 (2004). [10] G. B. Whitham, Linear and Nonlinear Waves (Wiley- ties. Interscience, New York, 1974). (3) Dislocation density jumps in single-slip. The for- [11] A. A. Benzerga, Y. Br´echet, A. Needleman, and E. Van mation of dislocation walls in the continuum theory can- der Giessen, Modeling and Simulation in Materials Sci- not properly be called a prediction, since wall formation ence 12, 159 (2004). [12] A. A. Benzerga, Y. Br´echet, A. Needleman, and E. Van is well known experimentally. It is, however, a natural der Giessen, Acta Materialia 53, 4765 (2005). consequence of the hyperbolic form of the equations. The [13] R. Fournet and J. M. Salazar, Physical Review B 53, (hitherto unobserved) prediction of dislocation density 6283 (1996). jumps in early stages of plasticity when only one slip sys- [14] D. G´omez-Garc´ia, B. Devincre, and L. Kubin, Physical tem is active is also a natural consequence of hyperbolic Review Letters 96, 125503 (2006). equations. Our continuum model is guaranteed to lower [15] I. Groma and B. Bak´o, Phys. Rev. Lett. 84, 1487 (2000). the net energy with time, reassuring us that the predicted [16] I. Groma and G. S. Pawley, Philos. Mag. A 67, 1459 dislocation density jumps are energetically favorable and (1993). [17] I. Groma and G. S. Pawley, Mater. Sci. Eng. A 164, 306 satisfy all compatibility constraints. They could, how- (1993). ever, be smeared by pinning and inhomogeneities in real [18] A. N. Gullouglu and C. S. Hartly, Model. Simul. Mater. systems just as grain boundaries and cell walls are dis- Sci. Eng. 1, 383 (1993). torted by these effects. A smeared dislocation density [19] D. B. Barts and A. E. Carlsson, Philosophical Magazine jump may be more challenging to identify than a smeared A 75, 541 (1997). wall of dislocations, perhaps explaining why these jumps [20] B. Bak´oand I. Groma, Phys. Rev. B 60, 122 (1999). have not yet been seen experimentally. [21] M. C. Miguel, A. Vespignani, S. Zapperi, J. Weiss, and J.-R. Grasso, Nature 410, 667 (2001). We thus predict that residual stress is not due to inho- [22] Our proposed evolution law equation 3 suppressed climb mogeneities or other infelicities in the formation process, by removing the trace of the current, hence the term − P but is an intrinsic component of the formation of grain λ/3 Jkkδij in ∂tβij . This choice is inconvenient here, P P boundaries and cell walls. We make concrete predictions because it introduces new components βyy and βzz to about the nature and form of these stresses near grain the problem, and the corresponding dislocation densities boundaries, that should be testable in colloidal systems ρzy, ρxz, and ρyz. While these are allowed by symmetry, they are not part of the discrete dislocation simulation. or using next-generation X-ray sources. We provide a [23] One concern we have is that a wall with a stress jump mesoscale explanation for one of the key morphological can lower its energy by splitting in two; stress jumps at distinctions between early (stage I) and later (stage III) walls in real materials may be stabilized by a different plasticity. Finally, we predict a new dislocation density mechanism than in the continuum model. 5

[24] The glide-only continuum model does have small jumps in the energy density.