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Shocks and slip systems: predictions from a theory of continuum dislocation dynamics Surachate Limkumnerd∗ and James P. Sethna† Laboratory of Atomic and Solid State Physics, Clark Hall, Cornell University, Ithaca, NY 14853-2501, USA (Dated: July 14, 2018) Using a recently developed continuum theory of dislocation dynamics, we derive three new predic- tions about plasticity and grain boundary formation in crystals. (1) There will be a residual stress jump across grain boundaries and plasticity-induced cell walls, which self-consistently acts to form the boundary by attracting neighboring dislocations; we derive the asymptotic late-time dynamics of the grain-boundary formation process. (2) At grain boundaries formed at high temperatures, there will be a cusp in the elastic energy density. (3) In early stages of plasticity, when only one type of dislocation is active (single-slip stage I plasticity), cell walls will not form; instead we predict the formation of jump singularities in the dislocation density. PACS numbers: 46.,91.60.Dc,91.60.Ed,47.40.-x Dislocation motion governs the plastic deformation of mapping the dislocation dynamics theory in one dimen- crystals, as well as the formation and evolution of grain sion onto Burgers equation. We will present numeri- boundaries in polycrystals. Acharya and Roy [1] recently cal simulations to flesh out these predictions. Finally, proposed an evolution law for the density of dislocations we will consider how and whether these predictions are in crystals, allowing both glide and climb (hence appro- likely to depend upon the detailed structure of the contin- priate at high temperatures). We subsequently rederived uum model, and how physical mechanisms ignored by the this law from the microscopic dynamics and a closure ap- model will likely affect and modify these striking predic- proximation, proposed a modified law to suppress climb, tions. We begin by introducing our notation and Acharya and showed numerically that both laws developed shock and Roy’s derivation of the equations of motion we study. singularities [2]. We argued that these shocks are the A complete macroscopic description of the deformation fundamental origin of the wall-like structures sponta- u E P E of a material is given by ∂iuj = βij + βij , where βij neously formed by dislocations—providing a continuum represents the elastic, reversible distortion and the plastic explanation of both the grain boundaries formed at high P distortion tensor βij describes the irreversible plastic de- temperatures and the cell walls formed at low temper- formation. The plastic distortion is the result of the net atures in plastically deformed crystals. Wall formation density of dislocations, described by the Nye dislocation thus emerges naturally from this continuum formulation. density tensor [3, 4, 5, 6, 7] What other properties and predictions naturally emerge from this mesoscale perspective? Here we discuss three. P α α α (1) Residual stress. The continuum model predicts ρij (x)= εilm∂lβ = t b δ(ξ ) (1) − mj i j residual stresses associated with the formation of grain Xα boundaries and cell walls. It makes detailed predictions about the asymptotic behavior of the stress and dislo- which measures the net flux of dislocation α, tangent to cation density near the walls, and about how the wall’s t, with Burgers vector b, in the (coarse-grained) neigh- associated stress allows it to draw in and absorb neigh- borhood of x. boring dislocations. The time evolution of the plastic distortion tensor is (2) Cusps in energy density. The continuum model P given in terms of a function Jij = ∂tβij , where εilmJmj predicts that the elastic energy associated with the resid- gives the net current for the conserved Burgers vector of ual stress is continuous across a grain boundary (despite the dislocation density tensor, ∂tρij = εilm∂lJmj . The the jumps in the individual stress components). The en- current from a single dislocation moving− with velocity v is arXiv:cond-mat/0610641v1 [cond-mat.dis-nn] 23 Oct 2006 ergy density does have a cusp singularity. Jij = εilntlbj vnδ(ξ), and the net Peach–Koehler force on (3) Dislocation density jumps in single-slip. When ap- PK a dislocation driving its motion is fl = εlmntmbcσnc propriately restricted to forbid climb, the model predicts where σ is the local stress (due, for example,− to the other that materials with only one kind of dislocation will not dislocations). In our derivation of the equation of mo- form cell walls, explaining the lack of such structures in tion [2], we allowed each dislocation to move indepen- systems (discrete-dislocation simulations and experimen- dently, and then made a closure approximation to write tal stage I plasticity) where only one slip system is active. J in terms of ρ. Roy and Acharya [1] got the same final Instead, the model predicts that these systems will form result by simply assuming that all the dislocations move a more subtle jump singularity in the dislocation density. with the same velocity v, given by a constant (D(ρ)/2) We will derive these three predictions analytically by times the force density l = εlmnρmcσnc on the local F − 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.

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