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On the role of dislocation conservation in single-slip crystal plasticity
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2011 Modelling Simul. Mater. Sci. Eng. 19 085002
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Modelling Simul. Mater. Sci. Eng. 19 (2011) 085002 (24pp) doi:10.1088/0965-0393/19/8/085002
On the role of dislocation conservation in single-slipcrystal plasticity
C B Hirschberger1,2, R H J Peerlings1, W A M Brekelmans1 andM G D Geers1
1 Eindhoven University of Technology, Mechanics of Materials Group, Materials TechnologyInstitute, PO Box 513, 5600 MB Eindhoven, The Netherlands2 Leibniz Universitat Hannover, Institute of Continuum Mechanics, Appelstr. 11, 30167Hannover, Germany
E-mail: [email protected], [email protected], [email protected] [email protected]
Received 18 November 2010, in final form 4 August 2011Published 16 September 2011Online at stacks.iop.org/MSMSE/19/085002
AbstractThe objective of this paper is to evaluate the relevance of dislocationconservation within the context of dislocation-based crystal plasticity. Inadvanced crystal plasticity approaches, dislocations play a prominent role.Their presence, nucleation, motion, and interactions enable the explanationand description of physical phenomena such as plastic slip, hardening andsize effects. While the conceptual aspects of the evolution and mechanicalconsequences of dislocations are treated analogously in a wide range ofadvanced crystal plasticity formulations, these formulations differ significantlywith respect to the underlying dislocation conservation properties. This paperidentifies and compares two essentially different approaches to model plasticdeformations in a single crystal. Both approaches have in common that theyrely on the geometrical relation between the plastic slip and the densities ofgeometrically necessary dislocations. In the first approach, the geometricalrelation serves as a balance and is supplemented with an evolution law for thestatistically stored dislocations. In the second approach the local conservationof the total number of dislocations is enforced in addition to the first balanceinstead of the evolution. Considering a single-slip model pile-up problem, thetwo representative frameworks are elaborated and confronted theoretically andnumerically on the basis of a dimensionless finite-element analysis to evaluatethe intrinsic role of dislocation conservation for model predictions.
(Some figures in this article are in colour only in the electronic version)
0965-0393/11/085002+24$33.00 © 2011 IOP Publishing Ltd Printed in the UK & the USA 1
Modelling Simul. Mater. Sci. Eng. 19 (2011) 085002 C B Hirschberger et al
1. Introduction
It is well accepted that plasticity-driven size effects may occur in stress–deformation relationsof miniaturized polycrystalline structures, in which the specimens consist of only a few grainsacross the thickness. These can be attributed to a nonuniform deformation field accompanied bya field of (geometrically necessary) dislocations and heterogeneities present in the polycrystal,as those influence or eventually obstruct the motion of dislocations, which carry the plastic slip.Among different types of heterogeneities (such as grain boundaries, second-phase particles orinclusions) rigid obstacles that are impenetrable to dislocations have the greatest influence onthe dislocation motion. As dislocations move towards such a rigid obstacle, dislocations cannotpenetrate due to the interatomic forces and the resulting mutual repulsion of dislocations of thesame orientation: a pile-up of dislocations against the boundary results [1]. Studies on suchpile-up situations are an important step in the understanding of the behaviour of dislocations atimpenetrable and eventually semi-penetrable boundaries such as grain boundaries. This is oneof the reasons that pile-up analyses have been performed with discrete dislocations (e.g. [2–4]).From such discrete studies, densities of dislocations can be evaluated providing the basis forfield theories of dislocations as they are used in advanced frameworks of crystal plasticity.To achieve meaningful predictions including the above-mentioned size effects, the mutualinteractions between the dislocations need to be properly accounted for in crystal plasticitymodels.
Rather than at the scale of individual dislocations, crystal plasticity frameworks modelplastic deformation in crystalline materials at the scale of individual crystals in a crystallite.Thereby the plastic deformation is obtained by a summation of the respective slip contributionson the distinct glide planes within the crystalline lattice. Beyond standard formulations ofcrystal plasticity, a multitude of higher-order or strain gradient crystal plasticity theorieshave been introduced to account for size effects stemming from dislocation interactions. Insuch advanced crystal plasticity formulations, the interaction between dislocations is capturedin a back stress term, containing a higher gradient of slip or—analogously—-a gradient ofcertain dislocation densities. We focus on those, which deal with additional balance equationsand consequently involve additional physical field variables to capture the nonlocal effectsoriginating from the dislocation interactions. These multifield theories [5] are denoted asformulations with external plastic variables in [6]. Among them we identify two majorcategories that—based on the respective assumptions—require one or two balance equationsfor the dislocation fields, as will be clarified next.
Crystal plasticity approaches with one dislocation balance. In a first category of approaches,the influence of the dislocations is captured based on the defect kinematics in the crystallinelattice. In this context, recall the notion of geometrically necessary dislocations (GNDs) thatwas first introduced by [7], describing the polarity of a dislocation field. These GNDs arerelated to lattice curvature [8, 9] and consequently to the gradient of slip in the crystal. Thegeometrical relation between the density and the slip gradient directly provides a dislocationbalance equation for higher-order crystal plasticity approaches. The time evolution of thedislocation population is described by evolution laws. Typical examples for this categoryof dislocation-based plasticity are e.g. the contributions of the group of Geers [10–12] aswell as [13]. As shown by Kuroda and Tvergaard [14] and by Erturk et al [15], these arelargely equivalent to the thermodynamically motivated gradient crystal plasticity frameworkproposed by Gurtin [16–19] formulated in terms of plastic slip and gradients thereof. Manycontributions including the thermodynamically consistent work of Svendsen [20], the papersof Ekh et al [13, 21] as well as the nonconvex extension by Yalcinkaya et al [22], fall in this first
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Modelling Simul. Mater. Sci. Eng. 19 (2011) 085002 C B Hirschberger et al
category. In addition, micromorphic or Cosserat types of crystal plasticity such as presentedby Forest [23–25] or by Bammann [26] are closely related to this first category as well.
Crystal plasticity approaches with two dislocation balances. While the vast majority ofdislocation-based crystal plasticity formulations pursues the previous type of approach, someworks follow a dislocation framework based on statistical mechanics [27]. This secondcategory of dislocation-based crystal plasticity shares the kinematically motivated balanceequation of the previous one. However, based on statistical physics considerations, theyadditionally postulate the local conservation of dislocations as long as no dislocation sources orsinks are active [27]. This conservation relation renders a second dislocation balance equationinstead an extra evolution equation. Recent applications of this second approach of crystalplasticity can be found e.g. in the papers of Arsenlis and Parks [28] or Yefimov et al [29, 30].A similar set of equations may be constructed on the basis of the classical continuum theoryof dislocations, see e.g. Sedlacek et al [31] or the field dislocation mechanics approach ofAcharya [32–34].
Objective and outline of this paper. This paper focuses on the role of the second balanceequation prescribing a conservation of dislocations for crystal plasticity and what the lackof such balance—as a characteristic of the more widely used first-mentioned category ofapproaches—implies for the predictions of dislocation evolution in dislocation pile up. The aimof this paper is not to develop a new theory, but rather to compare two widely used frameworksof dislocation-based crystal plasticity by considering a test case reduced to essential featuresneglecting multiple slip, interactions between slip planes, cross-slip, dislocation climb, etc.
To this end, we outline the two above-mentioned categories of dislocation-based crystalplasticity in more detail. Thereby we tend to make both categories as identical as possible,i.e. by taking the same formulation for the dislocation motion and back stress. Evidently, thisdeviates from some of the original propositions, but it provides an absolutely comparable setof equations for both approaches. For the sake of clarity, we choose a representative idealizedinitial-boundary value problem, which simulates the motion of dislocations within a channelenclosed by dislocation-impenetrable boundaries into a pile-up. The pile-up model problemidealizes plasticity within a single crystal under the assumption that the grain boundariesare impenetrable to dislocations. Following well-established examples in the literature[11, 17, 35–37] we consider the benchmark case of a constant stress field arising in a constrainedchannel under simple shear, which naturally satisfies macroscopic equilibrium in the sense thatdiv σ = 0 and can hence drop the dependence on the displacement field u here. Moreover,complications such as curved slip system are left out of consideration. A small strain settingis adopted for simplicity. For a single-slip system, a one-dimensional framework allows usto simulate the redistribution of dislocations predicted for a given set of dislocation balanceequations, see also [22, 38]. These simplified assumptions serve our purpose to focus on themere plastic process, i.e. the dislocation field problem and to identify the key characteristicsstemming from the nature of the respective governing equations.
Extensions of this idealized case to a fully coupled elasto-crystal-plastic problem involvingmultiple slip systems are part of future work. The main emphasis of this work is on the properunderstanding of the governing equations in a simple, one-dimensional setting.
2. Dislocation balance frameworks in crystal plasticity
In the following, to study the relevance of dislocation conservation, we extract the essentialsets of equations from the two different advanced crystal plasticity frameworks. Based on
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Modelling Simul. Mater. Sci. Eng. 19 (2011) 085002 C B Hirschberger et al
the relevant measures of dislocation densities and their corresponding kinematical properties,the particular assumptions on dislocation conservation and the resulting different sets ofdislocation balances are examined. Finally, the manner in which dislocation sources andsinks are accounted for is described.
2.1. Dislocation densities
In dislocation-based crystal plasticity, the dislocation distribution is kept track of using densitiesof dislocations on the respective glide planes. In particular, the total density of dislocations ona particular glide plane is defined as the total length of dislocation line per unit crystal volume.For straight edge dislocations with a line perpendicular to the considered surface this definitioncan be interpreted as the number of dislocation lines n piercing a local surface element. Thetotal dislocation density field, defined as a function ρ(x, t) of the position x on the glide planeand the time t , is conventionally decomposed as
ρ = ρS + |ρG| (1)
into the density of statistically stored dislocations (SSDs) and that of GNDs. While the first,ρS(x, t), is a positive (unsigned) field quantity, the GND density ρG(x, t) is a signed fieldquantity providing a measure for the polarity of the dislocation distribution. The GND densityis directly related to the lattice curvature [7, 8]. For the case of a single-slip system the GNDdensity can be expressed in terms of the gradient of slip
ρG = −1
b
∂γ
∂x, (2)
where b denotes the length of the Burgers vector.
2.2. Motion of dislocations
As typical for rate-dependent crystal plasticity formulations, the dislocations are assumed tobe mobile throughout the deformation process. This allows for a viscous-type drag relationbetween the aggregate dislocation velocity field v and the driving force on dislocations F ,which is also known as the Peach–Koehler force [1]:
v(x, t) = F
B. (3)
Herein the parameter B denotes a drag coefficient characterizing the thermally originatedresistance to dislocation motion through the lattice [39, 40]. This driving force on thedislocations comprises both an elastic stress (originating from long range elastic fields, e.g.due to external loading) and a back stress term (originating from short-range dislocationinteractions), multiplied by the Burgers vector length b:
F(x, t) = b [σ − σb] . (4)
The back stress σb accounts for the discrete interaction between dislocations, e.g. resultingfrom a gradient in the GND density. Here, we adopt a straightforward back stress formulation,
σb = G
2π [1 − ν]
b
ρ
∂ρG
∂x, (5)
for both following formulations, whereby G is the shear modulus and ν Poisson’s ratio.This particular formulation developed by [27, 36] based on arguments of statistical mechanicsaccounts exclusively for nearest neighbour interactions. Note that in addition to this choice,
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Modelling Simul. Mater. Sci. Eng. 19 (2011) 085002 C B Hirschberger et al
there are other models that capture the dislocation interaction within a certain cut-off radius([11, 12] etc). Furthermore, we here adopt the classical Orowan relation for the slip rate γ ,
γ = ρbv. (6)
Note that more elaborate flow rules [11, 41–43], incorporate additional features such as thermalactivation factors and an evolving slip resistance. They are omitted here, to restrict the paperto the essence of the comparative analysis.
2.3. Balance equations
The conservation of dislocations is treated differently in the two frameworks, yielding eitherone or two dislocation balances in the form of partial differential equations.
2.3.1. Framework 1. In the first category, ‘framework 1’, the influence of the dislocations isonly captured through the kinematic relation of the lattice curvature, (2). The rate form thereofwith the Orowan relation (6) constitutes the balance equation for the GNDs:
∂ρG
∂t+
∂
∂x(ρ v) = 0. (7)
This equation implies the conservation of the total Burgers vector, which can be shown byintegration over the entire domain.
This differential equation accounts for the GND only, while the total number of dislocationsremains uncontrolled. Hence this framework relies on the spontaneous production andannihilation of dislocations even in the absence of a source or sink term. An evolution law forthe SSDs,
ρS = s, (8)
in terms of sources and sink terms closes the dual set of equations for this framework. Theprecise format of the source and sink term s will be treated later.
2.3.2. Framework 2. The second category (denoted by ‘framework 2’) uses two differentialequations as the dislocations balances. Additional to the GND balance (7), a balance of thetotal density of dislocations is formulated as
∂ρ
∂t+
∂
∂x(ρG v) = s, (9)
which has been proposed by Groma [27, 36] based on arguments of statistical mechanics. Inthe absence of a source or sink, i.e. s = 0, (9) guarantees the local conservation of the numberof dislocations.
With the coupled problem consisting of two partial differential equations, (7) and (9), theevolution of dislocations is captured intrinsically. Additionally, the nucleation (source) andannihilation (sink) of dislocations can be controlled with the source term on the right-handside of (9), which will be made explicit next.
2.4. Sources and sinks of dislocations
Sources and sinks of dislocations are incorporated in the spirit of [44] as follows:
s = 1
b
[A
√ρ − Yρ
] |γ | . (10)
The two frameworks incorporate this source/sink term differently: while in framework 1, itgoverns the evolution of SSDs (8), in framework 2 this term directly occupies the right-hand
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Table 1. Parameters used in the crystal plasticity frameworks 1 and 2.
Burgers vector length b
Drag coefficient B
Shear modulus G
Poisson’s ratio ν
‘Multiplication’ parameter A
‘Annihilation’ parameter Y
Table 2. Geometrical data in a pile-up problem.
Domain length L
Initial dislocation density ρ0
side of the second balance equation (9). For further details on the possible choice of theparameters A and Y in the distinct contributions, the reader is referred to [11] and referencescited therein.
3. Dimensionless formulation
In the previous section, we stated that both formulations require two key equations: a balanceand an evolution in case of framework 1, or two balance equations for framework 2. Bothframeworks, 1 and 2, have in common a number of material and geometry parameters, whichare summarized in tables 1 and 2. However, as the two crystal plasticity frameworks canbe applied to various materials and boundary value problems, we aim for a formulation thatdisplays the characteristic features of the model independent from a particular choice of materialparameters. To this end, we pursue a dimensionless analysis on the two frameworks to studyhow certain combinations of parameters influence the response predicted by either approach.
3.1. Dimensionless parameters
Relative, dimensionless quantities are introduced and are indicated below by a superscript �.In particular, the position in space is considered relative to a domain length L, i.e. x� = x
L, and
accordingly the velocity is related to a reference velocity v� = Bσb
v, where σ is the constantapplied stress value. Consequently, the time that a dislocation would need to propagate freelythrough the entire domain L at constant speed σb/B, yields the dimensionless time t� = σb
BLt .
Likewise, the dislocation densities are related to the initial total density of dislocations ρ0
as to obtain relative dislocation densities for the total, ρ� = ρ/ρ0, the statistically stored,ρ�
S = ρS/ρ0 , and GNDs, ρ�G = ρG/ρ0. With these dimensionless quantities at hand, all
governing equations of section 2 can be rewritten in a dimensionless format.
3.2. Governing dimensionless equations
The dimensionless velocity, combined with (3)–(5), gives the dimensionless drag relation
v� =[
1 − G
2π [1 − ν]
1
σ
b
L
1
ρ�
∂ρ�G
∂x�
], (11)
with the dimensionless ratio
C := G
2π [1 − ν]
1
σ
b
L(12)
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Modelling Simul. Mater. Sci. Eng. 19 (2011) 085002 C B Hirschberger et al
Table 3. Dimensionless parameters used in the dimensionless formulations of the crystal plasticityframeworks 1 and 2.
‘Back stress’ parameter C = G
2π [1 − ν]
1
σ
b
LSource parameter S+ = AL
√ρ0
Sink parameter S− = YLρ0
identified directly. The parameter C characterizes the contribution of the back stress relativeto that of the applied stress σ in the aggregate dislocation velocity.
3.2.1. Framework 1. With the dimensionless dislocation densities of section 3.1, the GNDbalance (7) is reformulated as
∂ρ�G
∂t�+
∂
∂x�
(ρ� v�
) = 0 (13)
and the dimensionless evolution (8) reads
∂ρ�S
∂t�= s�. (14)
3.2.2. Framework 2. The dislocation balance (9) together with the source (10) reads in itsdimensionless format
∂ρ�
∂t�+
∂
∂x�
(ρ�
G v�) = s� (15)
for framework 2. The dimensionless GND balance as given by (13) also holds here.
3.2.3. Dislocation source and sink. The dimensionless source and sink term s� in (14) and(15) can be derived from (10) according to
s� =[S+
√ρ� − S−ρ�
]|ρ� v�|, (16)
where a dimensionless quantity for the source and the sink is identified as S+ = AL√
ρ0 andS− = YLρ0, respectively. The first parameter, S+, governs the creation or multiplication ofdislocations (‘source’), whereas the second one, S−, represents the annihilation (‘sink’) ofdislocations.
The three dimensionless parameters fully describing the model problems for bothframeworks are specified in table 3.
3.3. Initial-boundary value problem of pile-up
The pile-up problem mentioned earlier and sketched in figure 1 is studied within adimensionless setting. Under the restriction to single slip, we consider a domain of unitlength 1, B = [0, 1], in which the two opposite boundaries ∂B = {0, 1} are impenetrable todislocations. Starting from an initially homogeneous configuration of dislocations withoutdislocation polarity
ρ�(x�, t� = 0) = ρ�S(x
�, t� = 0) = 1 for x� ∈ B, (17)
ρ�G(x�, t� = 0) = 0 for x� ∈ B , (18)
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Modelling Simul. Mater. Sci. Eng. 19 (2011) 085002 C B Hirschberger et al
impenetrable
edge dislocations
Figure 1. Sketch of the boundary value problem: in vertical direction infinite channel withimpenetrable lateral boundaries at x� = 0 and x� = 1 and one slip system.
subjected to an externally applied field stress σ in the indicated direction, positive dislocationswill move to the right and negative ones to the left. As dislocations approach the impenetrableboundaries, where the dislocation flux is zero,
v�(x�, t�) = 0 for x� ∈ ∂B, (19)
they are expected to pile up.This motion towards the two dislocation-impenetrable boundaries ceases once an
equilibrium state is reached. Starting from the initial distribution of dislocations (17) and(18), dislocations redistribute and may change their contribution to the SSD and GND fieldquantities, as a result of the constant applied stress.
3.4. Numerical implementation
The numerical implementation using the finite-element method is outlined below and detailedin appendix A for both frameworks in terms of the dimensionless formulation of section 3.2.For the sake of clarity, we drop the � indicating dimensionless quantities below.
3.4.1. Framework 1. When using framework 1, see section 3.2.1, the dimensionless GNDdensity, ρG, can be used as the single nodal degree of freedom to be solved for3. The evolutionequation (14) requires a local numerical evaluation of the current SSD density, ρS. For detailson the numerical framework, reference is made to appendices A.1 and A.2.1.
3.4.2. Framework 2. For numerically solving the initial-boundary value problem offramework 2, two types of nodal unknowns are necessary. Following our theoretical treatmentof section 3.2.2, we here select both the dimensionless GND density ρG and the total dislocationdensity ρ to solve a fully coupled system of equations4. Further details on the numericalelaboration can be found in appendices A.1 and A.2.2.
3 Note that in connection with the full elasto-plastic crystal plasticity problem, thus two types of unknowns areneeded, i.e. e.g. the displacement u and ρα
G on each glide plane α, embedded into a coupled system of equations.4 For full elasto-plastic crystal plasticity problems, framework 2 entails three types of nodal unknowns, i.e. e.g. thedisplacement u, ρα
G and ρα on glide plane α. Hence the arising coupled system of equations is much larger than forthe previous framework.
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Modelling Simul. Mater. Sci. Eng. 19 (2011) 085002 C B Hirschberger et al
System of equations. Thus the balance equations (13) and (15) can be recast as two residualequations, which are discretized in and solved numerically.
3.4.3. Limitations of the chosen solution procedure. The balance prescribing the dislocationconservation (15) is an advection-dominated hyperbolic equation. Although standard Bubnov–Galerkin finite-element approaches are not the optimal solution approach, they are chosen inthis paper for the sake of simplicity. The constant C directly relates to the Peclet number,as Pe = 1/C, which is the ratio between advective and diffusion terms in the differentialequations (13) and (15) via the velocity (11). By choosing both the time step and the elementsize sufficiently small (here t = 5 × 10−4, le = 5 × 10−4) the element Peclet numberis controlled in order to avoid oscillations in the numerical solution. If the emphasis wereplaced on numerical solution methods, we needed to use for instance one of the followingmethodologies: streamline upwind/Petrov–Galerkin (SUPG) methods [45, 46], discontinousGalerkin methods [47–50], spurious oscillations at layers diminishing methods (SOLD) [51]or the variational multiscale method [52].
4. Numerical pile-up studies
In order to study the role of the dislocation conservation (9) in the pile-up prediction, wecompare both frameworks using the pile-up problem of dislocations sketched in section 3.3.To reveal the essential characteristics of the models, we first present numerical examplesomitting any source or sink. Then the influence of activated source and sink on the response isstudied. The differences in the predictions (and range of parameters in which either frameworkoperates well) are particularly pointed out.
As mentioned in section 3.4.3, the 1D domain is discretized with 2000 elements, and theadaptive time stepping scheme starts with an initial time step of t� = 5 × 10−4.
The spatial distributions of the dislocation densities, ρG, ρS and ρ, at an equilibrium stateare studied. To properly understand the characteristics of both models, the time evolution ofthe actual numbers of dislocations (nS, nG, n) in the entire domain is evaluated as well. Thesenumbers of dislocations and the plastic slip are simply evaluated through an integration ofthe fields over the spatial domain, for details see appendix B. In the following figures, unlessindicated otherwise, the plots on the left-hand side show the results for framework 1, while thecorresponding plots on the right-hand side present results for framework 2.
4.1. Absence of dislocation sources or sinks
In the first example, the essential behaviour of the two frameworks and in particular how theypredict the size effect arising from the dislocation pile-up is studied. To this end, only theparameter C defined in (12) is changed, while we omit any sink or source in (14) and (15), i.e.s = 0 (S+ = 0, S− = 0 in (16)).
4.1.1. Reference solutions. The first reference solutions are obtained for C = 1 in bothframeworks.
In framework 1, the SSDs cannot evolve for s = 0. Hence ρS remains constant overthe spatial domain (figure 2 left), and the number nS stays constant over time (figure 3 left).GNDs, however, are created in the process, which pile up against the dislocation-impenetrableboundaries (positive ρG on the right-hand side and negative ρG on the left-hand side; figure 2).As in framework 1 the SSD density remains constant for s = 0, only the GNDs contribute
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Modelling Simul. Mater. Sci. Eng. 19 (2011) 085002 C B Hirschberger et al
0 0.2 0.4 0.6 0.8 1–1
–0.5
0
0.5
1
1.5
2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1–1
–0.5
0
0.5
1
1.5
2
Figure 2. Reference solution for C = 1 (s = 0) for frameworks 1 (left) and 2 (right): spatialdistribution of dislocation densities at the equilibrium state.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1–1
–0.5
0
0.5
1
1.5
2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1–1
–0.5
0
0.5
1
1.5
2
Figure 3. Reference solution for C = 1 (s = 0) for frameworks 1 (left) and 2 (right): timeevolution of dislocation number towards the equilibrium state.
to an increase of the total number of dislocations, as is apparent in the time evolution of thenumber of dislocations (figure 3, left). The spatial distribution of the total dislocation densityρ has a kink at the symmetry axis of the chosen spatial domain.
The response obtained using framework 2 yields quantitatively similar results (figure 2and 3, right). The spatial GND density distribution exhibits a qualitatively comparable pile-upbehaviour. The main difference concerns the SSD density ρS, which reveals a kink at thesymmetry axis of the domain and decreases near the boundaries. The spatial distribution ofthe total dislocation density ρ remains smooth. The difference in framework 2 comparedwith framework 1 results from the fact that each GND is generated at the expense of an SSDby a spatial redistribution, since the total number of dislocations is preserved. This typicaldifference is best visible in figure 3.
The characteristic behaviour of the two frameworks, stemming from the properties oftheir respective sets of equations, is most pronounced at a lower value of C, e.g. for C = 0.1.Figure 4 shows that the pile-up characterized by the spatial distribution of GNDs, ρG, remainsqualitatively similar. However, with the axes chosen significantly differently, there are hugequantitative differences, which are also obvious in the time evolution of the number ofdislocations in figure 5. Recall that in framework 1, the number of SSDs governed by theevolution equation (14) is constant. With (13) as the only partial differential equation, thisframework does not constrain the total number of dislocations. Hence both the number ofGNDs and the total number of dislocations can grow considerably, whereby the time neededto reach equilibrium may be huge. Contrarily, framework 2, in which the total number ofdislocations is preserved, has the (logical) property that for smaller C (e.g. through a higher
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Modelling Simul. Mater. Sci. Eng. 19 (2011) 085002 C B Hirschberger et al
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1–150
–100
–50
0
50
100
150
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1–5
–4
–3
–2
–1
0
1
2
3
4
5
Figure 4. Reference solution for C = 0.1 (s = 0) for both frameworks 1 (left) and 2 (right):spatial distribution of dislocation densities at the equilibrium state.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1–150
–100
–50
0
50
100
150
0 0.5 1 1.50
0.5
1
1.5
Figure 5. Reference solution for C = 0.1 (s = 0) for both framework 1 (left) and 2 (right): timeevolution of dislocation number towards the equilibrium state.
stress σ ) a larger fraction of the dislocations is transformed from SSD to GND. The equilibriumstate (no dislocation motion) is reached much earlier than in framework 1.
In framework 2, for even smaller values of C, i.e. C < 0.1, the population of dislocations,which initially were all statistically stored, is entirely transformed into GNDs. In this case,when no dislocations are left at the domain centre to carry plastic slip, the limit responsebecomes very stiff and a further application of slip would encounter an absolutely rigid response(in the absence of elasticity). This so-called starvation property is only possible in framework2, due to its strict dislocation conservation.
4.1.2. Influence of the ratio C on the pile-up prediction. Simulations of the pile-up problemare next shown for both frameworks using three different values of C, still in the absence ofa source term, i.e. s = 0. With the previous reference solutions in mind, the influence of theparameter C on the evolution and the spatial distribution of the GNDs is studied in particular.
In figures 6–8, the spatial distribution of the slip γ and the GND density ρG as well as thetime evolution of the number of GNDs, nG, are shown for both frameworks. The larger thevalue of C, the more rigid is the predicted response of both frameworks. This means that ina small domain (L↓ ⇒ C ↑), the dislocation repulsion is strong, which causes considerablyless plastic slip. As C decreases, the amount of plastic slip and, correspondingly, the predicteddensity and number of GNDs increases. This influence is much greater for framework 1 ascan be seen in figure 7.
Framework 2 better reflects the pile-up of dislocation, represented by the GND densityρG near the impenetrable boundaries. Furthermore, as the number of GNDs is intrinsically
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Modelling Simul. Mater. Sci. Eng. 19 (2011) 085002 C B Hirschberger et al
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
C=0.1
C=1
C=10
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
C=0.1
C=1
C=10
Figure 6. Spatial distribution of the slip γ at the equilibrium state for framework 1 (left) and 2(right) for difference values of C (s = 0).
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1–2
–1.5
–1
–0.5
0
0.5
1
1.5
2
C=0.1
C=1
C=10
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1–2
–1.5
–1
–0.5
0
0.5
1
1.5
2
C=0.1
C=1
C=10
Figure 7. Spatial distribution of ρG at the equilibrium state for framework 1 (left) and 2 (right) fordifference values of C (s = 0).
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.5
1
1.5
2
2.5
C=0.1
C=1
C=10
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.5
1
1.5
2
2.5
C=0.1
C=1
C=10
Figure 8. Time evolution of nG near the equilibrium state for framework 1 (left) and 2 (right) fordifferent values of C (s = 0).
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Modelling Simul. Mater. Sci. Eng. 19 (2011) 085002 C B Hirschberger et al
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1–2
–1
0
1
2
3
4
5
ρρGρS
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1–2
–1
0
1
2
3
4
5
ρρGρS
Figure 9. Dislocation densities versus spatial position for frameworks 1 (left, computation divergedat t = 0.24) and 2 (right, at equilibrium) with source only (C = 1, S+ = 1, S− = 0) .
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Figure 10. Number of dislocations versus time for frameworks 1 (left) and 2 (right) with sourceonly (C = 1, S+ = 1, S− = 0).
limited by the model (if s = 0), the dislocation redistribution quantitatively limits the densityand number of GNDs.
4.2. Influence of dislocation sources and sinks
Based on the elementary results of section 4.1, the influence of dislocation sources and sinksaccording to (16) is next investigated. To this end, the source parameter S+ and the sinkparameter S− in equations (14) and (15) are first activated individually, before combinationsof source and sink parameters are studied for both frameworks.
4.2.1. Dislocation sources only. Figures 9 and 10 show the results of the pile-up problem inthe presence of a dislocation source but no sink term, i.e. C = 1, S+ = 1, S− = 0. For thisset of parameters, both frameworks behave significantly different. While for framework 2,the source has a low impact on the evolution of the total dislocation number and density, itseffect is pronounced for framework 1. For the latter, which lacks the capability of SSDs beingtransformed back into GNDs, the source causes the SSD density to become increasingly large in
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Modelling Simul. Mater. Sci. Eng. 19 (2011) 085002 C B Hirschberger et al
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1–2
–1
0
1
2
3
4
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1–2
–1
0
1
2
3
4
Figure 11. Dislocation densities versus spatial positions for frameworks 1 (left) and 2 (right) withsink only (C = 1, S+ = 0, S− = 1).
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.5
1
1.5
2
2.5
3
3.5
4
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.5
1
1.5
2
2.5
3
3.5
4
Figure 12. Number of dislocations versus time t for frameworks 1 (left) and 2 (right) with sinkonly (C = 1, S+ = 0, S− = 1).
the centre of the domain, while the GND distribution is not much different from the simulationwithout the source shown in figure 2. This overprediction of the number of SSDs and thusthe overall number of dislocations is attributed to this framework’s ineptity to control the totalnumber of dislocations and hence the missing restriction on the resulting SSD density, whichevolves according to (14). It even causes the model to fail in converging to an equilibriumstate, which becomes particularly obvious for the current parameter set in the time evolutionin figure 10 (left), in which the computation runs no further than t = 0.25.
4.2.2. Dislocation sink only. In case only the dislocation sink term is activated (C = 1,S+ = 0, S− = 1), the picture changes completely. The two frameworks predict a response thatis both qualitatively and quantitatively similar, as shown in figures 11 and 12. One differenceis that for framework 1, the SSD density ρS is primarily reduced in the centre of the domain,inducing a smoother distribution of the total dislocation density ρ for framework 1 comparedwith the reference solution of figure 2. The reduction of the number of dislocations is strongerfor framework 2 than for framework 1. All simulations with only a sink reach equilibriumwithin one (dimensionless) time unit.
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Modelling Simul. Mater. Sci. Eng. 19 (2011) 085002 C B Hirschberger et al
Figure 13. Number of SSDs at t = 1 depending on source S+ and sink S− parameters, forframeworks 1 (left) and 2 (right). Note the different vertical axes used in the two diagrams.
4.2.3. Both dislocation source and sink. Different from the quantity C, the source andsink parameters S+ and S− act primarily on the number of SSDs, which mainly influencesframework 1. The interaction of sinks and sources is evaluated by considering the final numbersof SSDs and GNDs, which are plotted for certain combinations of S+ and S− in figures 13and 14.
Adequately balancing the source by a sufficiently large sink regularizes the simulation suchthat equilibrium is recovered and meaningful pile-up predictions are obtained. In contrast,for framework 2 the influences of a dislocations source and sink on the numbers of SSDsand GNDs are moderate, yet particularly small in direct comparison with framework 1, seefigures 13(right) and 14(right).
Table 4 evaluates the influence of various combinations of dislocation sources and sinks fordifferent values of the parameter C. The conservation of dislocations of framework 2 appearsto be a safe yet possibly conservative assumption, which is taken as a reference. Framework1 is considered to yield reasonable results if these are qualitatively and quantitativelysimilar to those of framework 2. For large C, i.e. small domains or a high applied stress,both frameworks render comparable predictions. However, the sensitivity of framework 1to sources may be problematic and makes this crystal plasticity framework particularlyquestionable in large domains (small C). The incorporation of a dislocation sink providesa regularization, which is more effective for larger values of C and that coincides withreality in which usually both dislocation nucleation/multiplication and annihilation are activeconcurrently.
5. Discussion
Both frameworks have proven to be suitable to predict the expected size-dependent pile-upbehaviour studied here on an academic single-slip situation. The size of the domains (or theintensity of the stress field), represented by the parameter C, determines the influence of thegradient of slip and thus the size effect in the predicted response, see section 4.1. The shapeof the pile-up of GNDs is qualitatively predicted similarly for both frameworks, being moreconfined to the dislocation-impenetrable boundaries for larger domains. Framework 1, whichrelies on a spontaneous production of GNDs instead of a conservation of the total number ofdislocations, tends to quantitatively overpredict the number of GNDs near the boundaries. With
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Modelling Simul. Mater. Sci. Eng. 19 (2011) 085002 C B Hirschberger et al
Figure 14. Number of GNDs at t = 1 depending on source S+ and sink S− parameters, forframeworks 1 (left) and 2 (right). Note the different vertical axes used in the two diagrams.
Table 4. Influence of source and sink parameters on the equivalency of both frameworks:
� : framework 1 predicts a significantly different response than framework 2;◦ : framework 1 predicts a quantitatively similar, but qualitatively differentresponse compared to framework 2;
�: framework 1 and framework 2’s predictions are largely equivalent.
S− S+ C = 0.1 C = 1 C = 10
0 1 � � �0.5 1 � ◦ �0 0 � ◦ �1 0.5 ◦ ◦ �1 0 ◦ � �
its additional balance equation prescribing the conservation of the number of dislocations (9)in the absence of source or sink terms, framework 2 has significantly different properties.Due to the limited availability of dislocations, it is natural that framework 2 predicts a stifferresponse, which in the ultimate case may lead to the dislocation starvation, see section 4.1.1.In certain parameter ranges, the differences between the two frameworks may be either minoror pronounced: they are hardly present for mild loading or rather small domains (large C),yet large for high loading or large domains, as pointed out in section 4.1. One may concludethat in the absence of a source/sink term the dislocation conservation is especially relevant forlarge domains, in which the repelling of the dislocations does not play such a dominant role.In its absence (i.e. for framework 1) the number of dislocations is strongly overpredicted. Thereference solution neglecting source or sink contributions gives a first indication that the twoframeworks yield quantitatively comparable results in small domains or for a strong appliedstress only.
When a dislocation source and sink is activated as done in section 4.2, mainly the evolutionof SSDs is affected. For sources only, the differences between the predicted responses of thetwo frameworks become even more significant than before. In this case framework 1 becomesunreliable as it fails to reach an equilibrium state. It was shown that only if an active sourceis compensated by a dislocation sink, framework 1 delivers reasonable results. Although the
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Modelling Simul. Mater. Sci. Eng. 19 (2011) 085002 C B Hirschberger et al
results of framework 1 seem problematic at first glance especially in the absence of sinks, insimulations of problems representing realistic situations, where both dislocation sources andsinks are always present, this drawback may be negligible.
6. Conclusion and outlook
In this paper we have studied the role of dislocation conservation for two types of dislocation-based crystal plasticity represented by a single-slip problem. To this end, we have considered asingle-slip case to confront two essentially different formulations of dislocation-based crystalplasticity. In the first type, the nonlocal influence of the dislocation interaction is capturedbased on the geometric concept introduced by Ashby [8], by a single differential equation. Forinstance, the models by Evers/Bayley et al [11, 12], Gurtin [17] or Ekh et al [13] belong tothis framework. Contrarily in the second type of frameworks [28, 30], the local conservationof dislocations is prescribed by another differential equation, which was established by meansof statistical mechanics by Groma [27] for edge dislocations.
Crystal plasticity formulations of the first type have the advantage that in a finite-element context they are computationally cheaper, since they involve only half the numberof nodal degrees of freedom compared with the other frameworks. We have shown that in acertain range of parameters (high stress, small domains, low source–sink ratio), which is bestillustrated in table 4, the first type of frameworks can appropriately be used. This provides anopportunity to perform advanced crystal plasticity simulations at relatively low computationalcost. Note that approaches according to the first framework, in the literature often rely onmore refined slip evolution laws, which comprise a rate sensitivity parameter and thermaleffects. These additional effects may also influence the predictions. However, we have shownthat for certain cases, it may be inevitable to employ crystal plasticity formulations of thesecond type.
With the appropriate framework of dislocation-based crystal plasticity at hand, onecan perform simulations in single crystal plasticity that model the relevant size effects.Nevertheless, the current descriptions still call for improvement. One goal of futurework is to obtain an improved formulation of the back stress, as the assumptions currentapproaches are based on may be to crude. In particular, the nearest-neighbour-interaction-based back stress formulation [27] adopted here or even the incorporation of a certain cut-off radius as in [11] contain restricting assumptions. For instance, the works [3, 4] haveprovided certain insight into the dislocation interactions and grounds for further improvement.Particularly for problems involving multiple slip systems, the mutual interactions betweenthe glide systems, known as cross hardening, are not understood. In multiple slip casesalso cross-slip can occur, which means that by dislocations changing their slip systems,the dislocation conservation mechanisms may be more complicated and not as strict asprescribed here by the second framework. This study is exclusively restricted to dislocationglide, which suffices for crystals with face-centred cubic (fcc) atom structure. Whendealing with crystals of body-centred cubic (bcc) structure [53, 54] additionally dislocationclimb needs to be taken into account. The simulation of pile up against dislocation-impenetrable boundaries serves as a limit case for dislocation interactions in generalheterogeneous crystallites which we find in multi-phase polycrystalline materials. Whereas theassumption may hold well for phase boundaries, experiments indicate that grain boundariesin polycrystals are rather semi-penetrable to dislocations. Criteria for dislocation motionacross grain boundaries and the modelling thereof is the subject of ongoing researchas e.g. [55–57].
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Modelling Simul. Mater. Sci. Eng. 19 (2011) 085002 C B Hirschberger et al
Table A1. Notation for the numerical quantities.
Finite-element quantitiesρT
J nodal degree of freedom of the total dislocation densityρG
L nodal degree of freedom of the GND densityN◦• shape functionsR◦
• residual vectorK◦
• tangent matrix
SuperscriptsT refers to total dislocation densityG refers to GND densityS refers to SSD density(k) iteration step in local Newton–Raphson loop
Subscripts(n) evaluated at the beginning of time step n
(n + 1) evaluated at the end of time step n
I , K node number of the test function for ρ or ρG, respectivelyJ , L node number of the trial function for ρ or ρG respectively
Appendix A. Numerical implementation
The finite-element procedure to solve the governing equations of both frameworks is outlinedin this section, using the notation indicated in table A1. Neglecting the complicationsmentioned in section 3.4, we straightforwardly discretize the time-dependent problems withfinite differences in time and Bubnov–Galerkin finite elements in space. As already pointedout in [29], one needs to be aware, however, that the presented solution procedure is not theoptimal choice to tackle the advection-dominated parabolic differential equations [58, 59]. Itrequires extremely small elements and very small time steps in order to ensure numericalstability and to circumvent numerical oscillations.
For both frameworks, the discrete residual, tangent matrix and solution algorithms arederived. When considering the discrete weak formulations of the dislocation balances, theintegrals of the residual vectors and the tangent matrices are evaluated elementwise by meansof numerical integration and assembled to the global matrices in a standard finite-elementmanner.
A.1. Approximation
The test and trial functions for ρG and ρ are denoted by (wG; ρG) and (wT; ρT), whereby thesuperscript T or G identify quantities associated with the total or the GND density, respectively.In a general case, the nodal shape functions N◦
• used to discretize the two sets of functions canbe chosen differently.
A.1.1. Approximation of the unknown dislocation densities. For both frameworks, the GNDdensity and the associated test functions are discretized as
ρG(x, t) =nG∑L
NGL (x) ρG
L (t), (A.1)
wG(x, t) =nG∑K
NGK(x) wG
K(t). (A.2)
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Modelling Simul. Mater. Sci. Eng. 19 (2011) 085002 C B Hirschberger et al
Likewise, the total dislocation density and the associated test functions approximations aregiven by
ρT(x, t) =nT∑J
NTJ (x) ρT
J (t) , (A.3)
wT(x, t) =nT∑I
NTI (x) wT
I (t) , (A.4)
which are only used in framework 2.
A.1.2. Approximation of the spatial derivatives of the dislocation densities. The derivativesof the GND density used in both frameworks and the density of the total number of dislocationsused in framework 2 are determined in a standard manner:
∂ρG
∂x(x, t) =
nG∑L
∂NGK
∂x(x) ρG
L (t), (A.5)
∂ρT
∂x(x, t) =
nT∑J
∂NTJ
∂x(x) ρT
J (t). (A.6)
A.2. Numerical solution of the initial-boundary value problem
The finite-element solution procedures for both frameworks are elaborated next. Based on thenumber of PDEs the first framework requires a local solution of the SSD evolution equation,while the second one entails a coupled finite-element system of equations.
A.2.1. Framework 1. With the GND density (A.1) as the nodal degree of freedom, initial-boundary value problems are solved using a finite-element discretization in space and finitedifferences in time as follows.
System of equations. The weak formulation of the local balance of GND densities (13),taking into account the boundary conditions (19), directly renders the residual for the GNDdiscretized both in time and space as
RGK(n+1) =
∫B
[NG
K
ρG(n+1) − ρG
(n)
t− dNG
K
dxρT
(n+1)v(n+1)
]dx
.= 0. (A.7)
Herein the boundary term stemming from d(wGρGv)/dx vanishes due to the dislocationobstacle (19) at both boundaries.
Linearization. In each time step, we obtain a solution iteratively using a Newton–Raphsonscheme. To this end, we linearize the residual (A.7) with respect to the nodal degree offreedom ρG
L :
RGK(n+1) + KGG
KL ρGL
.= 0. (A.8)
This system of equations is solved for the increments ρGL = ρG
L(n+1) −ρGL(n) until convergence
is reached. Herein the tangent operator KGGKL is determined from the derivative of the residual as
KGGKL = DρG
L(n+1)(RG
K(n+1))
=∫
B
[1
tNG
KNGL − sign (ρG
(n+1))dNG
K
dxNG
L + CdNG
K
dx
dNGL
dx
]dx. (A.9)
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Modelling Simul. Mater. Sci. Eng. 19 (2011) 085002 C B Hirschberger et al
In (A.7) and (A.9), the total dislocation density is determined from the sum (1). Theconvenient property
∂(ρTv)
∂ρT= v + ρT ∂v
∂ρT= 1 (A.10)
is exploited for the tangent (A.9) and in what follows. Both the SSD density and the dislocationvelocity are computed as described in the following.
Dislocation evolution. To compute the velocity and the total dislocation density in (A.7) and(A.9), at each integration point we need to determine the SSD density at the end of the nth timestep with the evolution equation (14). The SSD density rate (14) at the end of the nth time stepis approximated linearly in time using an implicit Euler time discretization. The local residualfunction then reads
rS = ρS(n+1) − ρS
(n) − s(n+1) t.= 0 (A.11)
with s(n+1) = ρS(n+1) from equation (16). Using a Newton–Raphson scheme, starting from the
initial value from the history field, we iteratively solve (A.11) according to
rS(k) +∂rS
∂ρS(n+1)
∣∣∣∣∣(k)
ρS(n+1) = 0 (A.12)
for the unknown update on the SSD density ρS(n+1) = ρ
S(k+1)
(n+1) − ρS(k)
(n+1) until convergenceis reached. Herein, with the SSD density rate given in (14), the tangent operator in the kthiteration reads
∂rS
∂ρS(n+1)
∣∣∣∣∣(k)
= 1 −[
1
2S+(ρ
T(k)
(n+1))− 1
2 − S−]
|ρT(k)
(n+1)v(k)
(n+1)|
−[S+
√ρ
T(k)
(n+1) − S−ρT(k)
(n+1)
]sign (v
(k)
(n+1)). (A.13)
Note that in each time step, ρG(n+1) is fixed, while the velocity
v(k)
(n+1) =[
1 − C1
ρT(k)
(n+1)
∂ρG(n+1)
∂x
](A.14)
and the total density ρT(k)
(n+1) = ρS(k)
(n+1) + |ρG(n+1)| are updated in each iteration.
A.2.2. Framework 2. For the finite-element solution of framework 2, both the total densityof dislocations (A.3) and the density of GNDs (A.1) are selected as finite-element degrees offreedom.
Residual. Hence, considering the dimensionless weak formulations of the balances (13) and(15), we obtain the residuals for ρT
J and ρGL , discretized both in time and space,
RTI (n+1) =
∫B
[NT
I
ρT(n+1) − ρT
(n)
t− dNT
I
dxρG
(n+1) v(n+1)
]dx
−∫
BNT
I
[S+
√ρT
(n+1) − S−ρT(n+1)
] ∣∣ρT(n+1) v(n+1)
∣∣ dx.= 0, (A.15)
RGK(n+1) =
∫B
[NG
K
ρG(n+1) − ρG
(n)
t− dNG
K
dxρT
(n+1) v(n+1)
]dx
.= 0, (A.16)
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Modelling Simul. Mater. Sci. Eng. 19 (2011) 085002 C B Hirschberger et al
which both have to vanish upon convergence in each time step. Note that as in (A.7), theboundary terms stemming from d(wTρTv)/dx and d(wGρGv)/dx vanish due to the boundarycondition (19) .
Linearization of the coupled problem. Employing a Newton–Raphson solution scheme again,the two residuals (A.15)–(A.16) are linearized with respect to both unknowns. This rendersthe following coupled system of equations[
RTI (n+1)
RGK(n+1)
]+
[KTT
IJ (n+1) KTGIL(n+1)
KGTKJ(n+1) KGG
KL(n+1)
] [ρT
J (n+1)
ρGL(n+1)
].=
[0I
0K
], (A.17)
which needs to be solved. Herein, the particular tangent matrices are obtained with the partialderivatives at the end of the time step as
KTTIJ = DρT
J(RT
I ) =∫
B
[1
tNT
I NTJ − C
ρG
(ρT)2
dρG
dx
dNTI
dxNT
J
]dx
−∫
BNT
I NTJ
{[1
2S+(ρT)−
12 − S−
]|ρTv| dx
+∫
B
[S+
√ρT − S−ρT
]sign (ρTv) dx,
KTGIL= DρG
L(RT
I ) = −∫
B
[dNT
I
dx
[NG
L v − CρG
ρT
dNGL
dx
]dx
+∫
B
[S+
√ρT − S−ρT
]C sign (ρTv)NT
I
dNGL
dxdx,
KGGKL= DρG
L
(RG
K
) =∫
B
[1
tNG
KNGL + C
dNGK
dx
dNGL
dx
]dx,
KGTKJ = DρG
J
(RT
K
) = −∫
B
dNGK
dxNT
J dx,
with all quantities evaluated at the end of the time step and omitting the subscripts (n + 1) herefor brevity. Herein, the velocity is given by
v(n+1) =[
1 − C1
ρT(n+1)
∂ρG(n+1)
∂x
], (A.18)
while its derivatives needed in the tangent matrices are determined as
∂v(n+1)
∂ρTJ (n+1)
= C1
(ρT(n+1))
2
∂ρG(n+1)
∂xNT
J , (A.19)
∂v(n+1)
∂ρGL(n+1)
= − C1
ρT(n+1)
dNGL
dx, (A.20)
which again employ the property (A.10).
Appendix B. Computation of discrete field quantities
B.1. Number of dislocations
The number of dislocations on the slip system within the entire domain can be obtained by anintegration over the respective densities
n(t) =∫
Bρ(x, t) dx, (B.1)
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Modelling Simul. Mater. Sci. Eng. 19 (2011) 085002 C B Hirschberger et al
nS(t) =∫
BρS(x, t) dx, (B.2)
nG(t) =∫
B|ρG(x, t)| dx (B.3)
with B = [0, 1]. For the discrete nodal quantities, the numbers can be approximated forinstance using a midpoint rule in each finite element.
B.2. Plastic slip
The plastic slip on the glide system can be recomputed by an integration of (2) with the nonzeroslip boundary condition, γ = 0, on the impenetrable boundaries.
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